![288 MATHEMATICS
There is a connection, known as the Fundamental Theorem of Calculus, between
indefinite integral and definite integral which makes the definite integral as a practical
toolforscienceandengineering.Thedefiniteintegralisalsousedtosolvemanyinteresting
problems from various disciplines like economics, finance and probability.
In this Chapter, we shall confine ourselves to the study of indefinite and definite
integrals and their elementary properties including some techniques of integration.
7.2 Integration as an Inverse Process of Differentiation
Integration is the inverse process of differentiation. Instead of differentiating a function,
we are given the derivative of a function and asked to find its primitive, i.e., the original
function. Such a process is called integration or anti differentiation.
Let us consider the following examples:
We know that (sin )
d
x
dx
= cos x ... (1)
3
( )
3
d x
dx
= x2 ... (2)
and ( )
x
d
e
dx
= ex ... (3)
We observe that in (1), the function cos x is the derived function of sin x. We say
that sin xis an anti derivative (or an integral) of cos x. Similarly, in (2) and (3),
3
3
x
and
ex
are the anti derivatives (or integrals) of x2
and ex
, respectively. Again, we note that
for any real number C, treated as constant function, its derivative is zero and hence, we
can write (1), (2) and (3) as follows :
(sin + C) cos
=
d
x x
dx
,
3
2
( + C)
3
=
d x
x
dx
and ( + C) =
x x
d
e e
dx
Thus, anti derivatives (or integrals) of the above cited functions are not unique.
Actually, there exist infinitely many anti derivatives of each of these functions which
can be obtained by choosing C arbitrarily from the set of real numbers. For this reason
C is customarily referred to as arbitrary constant. In fact, C is the parameter by
varying which one gets different anti derivatives (or integrals) of the given function.
More generally, if there is a function F such that F ( ) = ( )
d
x f x
dx
, ∀ x ∈ I (interval),
then for any arbitrary real number C, (also called constant of integration)
[ ]
F ( ) + C
d
x
dx
= f (x), x ∈ I](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-2-2048.jpg)
![294 MATHEMATICS
A Note The equivalence of the families { }
1 1
( ) + C ,C
f x dx ∈
∫ R and
{ }
2 2
( ) + C ,C
g x dx ∈
∫ R is customarily expressed by writing ( ) = ( )
f x dx g x dx
∫ ∫ ,
without mentioning the parameter.
(III) [ ]
( ) + ( ) ( ) + ( )
f x g x dx f x dx g x dx
=
∫ ∫ ∫
Proof By Property (I), we have
[ ( ) + ( )]
d
f x g x dx
dx
∫ = f (x) + g(x) ... (1)
On the otherhand, we find that
( ) + ( )
d
f x dx g x dx
dx
∫ ∫ = ( ) + ( )
d d
f x dx g x dx
dx dx
∫ ∫
= f (x) + g(x) ... (2)
Thus, in view of Property (II), it follows by (1) and (2) that
( )
( ) ( )
f x g x dx
+
∫ = ( ) ( )
f x dx g x dx
+
∫ ∫ .
(IV) For any real number k, ( ) ( )
k f x dx k f x dx
=
∫ ∫
Proof By the Property (I), ( ) ( )
d
k f x dx k f x
dx
=
∫ .
Also ( )
d
k f x dx
dx
∫ = ( ) = ( )
d
k f x dx k f x
dx ∫
Therefore, using the Property (II), we have ( ) ( )
k f x dx k f x dx
=
∫ ∫ .
(V) Properties (III) and (IV) can be generalised to a finite number of functions
f1
, f2
, ..., fn
and the real numbers, k1
, k2
, ..., kn
giving
[ ]
1 1 2 2
( ) ( ) ( )
n n
k f x k f x ... k f x dx
+ + +
∫
= 1 1 2 2
( ) ( ) ( )
n n
k f x dx k f x dx ... k f x dx
+ + +
∫ ∫ ∫ .
To find an anti derivative of a given function, we search intuitively for a function
whose derivative is the given function. The search for the requisite function for finding
an anti derivative is known as integration by the method of inspection. We illustrate it
through some examples.](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-8-2048.jpg)
![INTEGRALS 295
Example 1 Write an anti derivative for each of the following functions using the
method of inspection:
(i) cos 2x (ii) 3x2 + 4x3 (iii)
1
x
, x ≠ 0
Solution
(i) We look for a function whose derivative is cos 2x. Recall that
d
dx
sin 2x = 2 cos 2x
or cos 2x =
1
2
d
dx
(sin 2x) =
1
sin 2
2
d
x
dx
Therefore, an anti derivative of cos 2x is
1
sin 2
2
x .
(ii) We look for a function whose derivative is 3x2 + 4x3. Note that
( )
3 4
d
x x
dx
+ = 3x2
+ 4x3
.
Therefore, an anti derivative of 3x2
+ 4x3
is x3
+ x4
.
(iii) We know that
1 1 1
(log ) 0 and [log ( )] ( 1) 0
d d
x ,x – x – ,x
dx x dx – x x
= > = = <
Combining above, we get ( )
1
log 0
d
x , x
dx x
= ≠
Therefore,
1
log
dx x
x
=
∫ is one of the anti derivatives of
1
x
.
Example 2 Find the following integrals:
(i)
3
2
1
x –
dx
x
∫ (ii)
2
3
( 1)
x dx
+
∫ (iii) ∫
3
2
1
( 2 – )
+
∫
x
x e dx
x
Solution
(i) We have
3
2
2
1 –
x –
dx x dx – x dx
x
=
∫ ∫ ∫ (by Property V)](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-9-2048.jpg)
![298 MATHEMATICS
(iii) When the variable of integration is denoted by a variable other than x, the integral
formulae are modified accordingly. For instance
4 1
4 5
1
C C
4 1 5
y
y dy y
+
= + = +
+
∫
7.2.3 Comparison between differentiation and integration
1. Both are operations on functions.
2. Both satisfy the property of linearity, i.e.,
(i) [ ]
1 1 2 2 1 1 2 2
( ) ( ) ( ) ( )
d d d
k f x k f x k f x k f x
dx dx dx
+ = +
(ii) [ ]
1 1 2 2 1 1 2 2
( ) ( ) ( ) ( )
k f x k f x dx k f x dx k f x dx
+ = +
∫ ∫ ∫
Here k1 and k2 are constants.
3. Wehavealreadyseenthatallfunctionsarenotdifferentiable.Similarly, all functions
are not integrable. We will learn more about nondifferentiable functions and
nonintegrable functions in higher classes.
4. The derivative of a function, when it exists, is a unique function. The integral of
a function is not so. However, they are unique upto an additive constant, i.e., any
two integrals of a function differ by a constant.
5. When a polynomial function P is differentiated, the result is a polynomial whose
degree is 1 less than the degree of P.When a polynomial function P is integrated,
the result is a polynomial whose degree is 1 more than that of P.
6. We can speak of the derivative at a point. We never speak of the integral at a
point, we speak of the integral of a function over an interval on which the integral
is defined as will be seen in Section 7.7.
7. The derivative of a function has a geometrical meaning, namely, the slope of the
tangent to the corresponding curve at a point. Similarly, the indefinite integral of
a function represents geometrically, a family of curves placed parallel to each
other having parallel tangents at the points of intersection of the curves of the
familywiththelinesorthogonal(perpendicular)totheaxisrepresentingthevariable
of integration.
8. The derivative is used for finding some physical quantities like the velocity of a
moving particle, when the distance traversed at any time t is known. Similarly,
the integral is used in calculating the distance traversed when the velocity at time
t is known.
9. Differentiation is a process involving limits.So is integration, as will be seen in
Section 7.7.](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-12-2048.jpg)
![306 MATHEMATICS
Solution
(i) Recall the identity cos 2x = 2 cos2
x – 1, which gives
cos2
x =
1 cos 2
2
x
+
Therefore,
2
cos
∫ x dx =
1
(1+ cos 2 )
2
x dx
∫ =
1 1
cos 2
2 2
dx x dx
+
∫ ∫
=
1
sin 2 C
2 4
x
x
+ +
(ii) Recall the identity sin x cos y =
1
2
[sin (x + y) + sin (x – y)] (Why?)
Then sin 2 cos 3
∫ x xdx =
1
sin 5 sin
2
•
∫ ∫
x dx x dx
=
1 1
cos 5 cos C
2 5
– x x
+ +
=
1 1
cos 5 cos C
10 2
– x x
+ +
(iii) From the identity sin 3x = 3 sin x – 4 sin3 x, we find that
sin3
x =
3sin sin 3
4
x – x
Therefore,
3
sin x dx
∫ =
3 1
sin sin 3
4 4
x dx – x dx
∫ ∫
=
3 1
– cos cos 3 C
4 12
x x
+ +
Alternatively,
3 2
sin sin sin
x dx x x dx
=
∫ ∫ =
2
(1– cos ) sin
x x dx
∫
Put cos x = t so that – sin x dx = dt
Therefore,
3
sin x dx
∫ = ( )
2
1 – t dt
− ∫ =
3
2
C
3
t
– dt t dt – t
+ = + +
∫ ∫
=
3
1
cos cos C
3
– x x
+ +
Remark It can be shown using trigonometric identities that both answers are equivalent.](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-20-2048.jpg)
![308 MATHEMATICS
(2) ∫ 2 2
1 +
= log + C
2 –
–
dx a x
a a x
a x
(3) ∫
– 1
2 2
1
tan C
dx x
= +
a a
x + a
(4) ∫
2 2
2 2
= log + – +C
–
dx
x x a
x a
(5) ∫
– 1
2 2
= sin + C
–
dx x
a
a x
(6) ∫
2 2
2 2
= log + + + C
+
dx
x x a
x a
We now prove the above results:
(1) We have 2 2
1 1
( ) ( )
x – a x a
x – a
=
+
=
1 ( ) – ( ) 1 1 1
2 ( ) ( ) 2
x a x – a
–
a x – a x a a x – a x a
+
=
+ +
Therefore, 2 2
1
2
dx dx dx
–
a x – a x a
x – a
=
+
∫ ∫ ∫
= [ ]
1
log ( )| log ( )| C
2
| x – a – | x a
a
+ +
=
1
log C
2
x – a
a x a
+
+
(2) In view of (1) above, we have
2 2
1 1 ( ) ( )
2 ( ) ( )
–
a x a x
a a x a x
a x
+ + −
=
+ −
=
1 1 1
2a a x a x
+
− +
](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-22-2048.jpg)
![INTEGRALS 309
Therefore, 2 2
–
dx
a x
∫ =
1
2
dx dx
a a x a x
+
− +
∫ ∫
=
1
[ log | | log | |] C
2
a x a x
a
− − + + +
=
1
log C
2
a x
a a x
+
+
−
ANote The technique used in (1) will be explained in Section 7.5.
(3) Put x = a tan θ. Then dx = a sec2
θ dθ.
Therefore, 2 2
dx
x a
+
∫ =
2
2 2 2
θ θ
θ
sec
tan
a d
a a
+
∫
=
1
1 1 1
θ θ C tan C
– x
d
a a a a
= + = +
∫
(4) Let x = a secθ. Then dx = a secθ tanθ dθ.
Therefore, 2 2
dx
x a
−
∫ = 2 2 2
secθ tanθ θ
sec θ
a d
a a
−
∫
= 1
secθ θ log secθ + tanθ + C
d =
∫
=
2
1
2
log 1 C
x x
–
a a
+ +
=
2 2
1
log log C
x x – a a
+ − +
=
2 2
log + C
x x – a
+ , whereC = C1
– log|a|
(5) Let x = a sinθ. Then dx = a cosθ dθ.
Therefore, 2 2
dx
a x
−
∫ = 2 2 2
θ θ
θ
cos
sin
a d
a – a
∫
=
1
θ = θ + C = sin C
– x
d
a
+
∫
(6) Let x = a tanθ. Then dx = a sec2θ dθ.
Therefore,
2 2
dx
x a
+
∫ =
2
2 2 2
θ θ
θ
sec
tan
a d
a a
+
∫
= 1
θ θ
secθ θ = log (sec tan ) C
d + +
∫](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-23-2048.jpg)
![INTEGRALS 311
(10) For the evaluation of the integral of the type
2
( )
px q dx
ax bx c
+
+ +
∫ , we proceed
as in (9) and transform the integral into known standard forms.
Let us illustrate the above methods by some examples.
Example 8 Find the following integrals:
(i) 2
16
dx
x −
∫ (ii) 2
2
dx
x x
−
∫
Solution
(i) We have 2 2 2
16 4
dx dx
x x –
=
−
∫ ∫ =
4
log C
8 4
x –
x
1
+
+
[by 7.4 (1)]
(ii)
( )
2 2
2 1 1
=
−
∫ ∫
dx dx
x x – x –
Put x – 1 = t. Then dx = dt.
Therefore, 2
2
dx
x x
−
∫ = 2
1
dt
– t
∫ =
1
sin ( ) C
–
t + [by 7.4 (5)]
=
1
sin ( –1) C
–
x +
Example 9 Find the following integrals :
(i) 2
6 13
dx
x x
− +
∫ (ii) 2
3 13 10
dx
x x
+ −
∫ (iii) 2
5 2
dx
x x
−
∫
Solution
(i) We have x2 – 6x + 13 = x2 – 6x + 32 – 32 + 13 = (x – 3)2 + 4
So,
6 13
dx
x x
2
− +
∫ =
( )2 2
1
3 2
dx
x – +
∫
Let x – 3 = t. Then dx = dt
Therefore,
6 13
dx
x x
2
− +
∫ =
1
2 2
1
tan C
2 2
2
–
dt t
t
= +
+
∫ [by 7.4 (3)]
=
1
1 3
tan C
2 2
– x –
+](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-25-2048.jpg)
![312 MATHEMATICS
(ii) The given integral is of the form 7.4 (7). We write the denominator of the integrand,
2
3 13 10
x x –
+ =
2 13 10
3
3 3
x
x –
+
=
2 2
13 17
3
6 6
x –
+
(completing the square)
Thus
3 13 10
dx
x x
2
+ −
∫ = 2 2
1
3 13 17
6 6
dx
x
+ −
∫
Put
13
6
x t
+ = . Then dx = dt.
Therefore,
3 13 10
dx
x x
2
+ −
∫ = 2
2
1
3 17
6
dt
t
−
∫
= 1
17
1 6
log C
17 17
3 2
6 6
t –
t
+
× × +
[by 7.4 (i)]
= 1
13 17
1 6 6
log C
13 17
17
6 6
x –
x
+
+
+ +
= 1
1 6 4
log C
17 6 30
x
x
−
+
+
= 1
1 3 2 1 1
log C log
17 5 17 3
x
x
−
+ +
+
=
1 3 2
log C
17 5
x
x
−
+
+
, where C = 1
1 1
C log
17 3
+](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-26-2048.jpg)
![INTEGRALS 313
(iii) We have
2 2
5 2 5
5
dx dx
x
x x x –
2
=
−
∫ ∫
= 2 2
1
5 1 1
5 5
dx
x – –
∫ (completing the square)
Put
1
5
x – t
= . Then dx = dt.
Therefore,
5 2
dx
x x
2
−
∫ = 2
2
1
5 1
5
dt
t –
∫
=
2
2
1 1
log C
5
5
t t –
+ +
[by 7.4 (4)]
=
2
1 1 2
log C
5 5
5
x
x – x –
+ +
Example 10 Find the following integrals:
(i)
2
2 6 5
x
dx
x x
2
+
+ +
∫ (ii) 2
3
5 4
x
dx
x x
+
− +
∫
Solution
(i) Using the formula 7.4 (9), we express
x + 2 = ( )
2
A 2 6 5 B
d
x x
dx
+ + + = A (4 6) B
x + +
Equating the coefficients of x and the constant terms from both sides, we get
4A = 1 and 6A + B = 2 or A =
1
4
and B =
1
2
.
Therefore,
2
2 6 5
x
x x
2
+
+ +
∫ =
1 4 6 1
4 2
2 6 5 2 6 5
x dx
dx
x x x x
2 2
+
+
+ + + +
∫ ∫
= 1 2
1 1
I I
4 2
+ (say) ... (1)](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-27-2048.jpg)
![314 MATHEMATICS
In I1, put 2x2
+ 6x + 5 = t, so that (4x + 6) dx = dt
Therefore, I1
= 1
log C
dt
t
t
= +
∫
=
2
1
log | 2 6 5| C
x x
+ + + ... (2)
and I2
= 2
2
1
5
2
2 6 5 3
2
dx dx
x x x x
=
+ + + +
∫ ∫
= 2 2
1
2 3 1
2 2
dx
x
+ +
∫
Put
3
2
x t
+ = , so that dx = dt, we get
I2
= 2
2
1
2 1
2
dt
t
+
∫ =
1
2
1
tan 2 C
1
2
2
–
t +
×
[by 7.4 (3)]
=
1
2
3
tan 2 + C
2
–
x
+
= ( )
1
2
tan 2 3 + C
–
x + ... (3)
Using (2) and (3) in (1), we get
( )
2 1
2 1 1
log 2 6 5 tan 2 3 C
4 2
2 6 5
–
x
dx x x x
x x
2
+
= + + + + +
+ +
∫
where, C = 1 2
C C
4 2
+
(ii) This integral is of the form given in 7.4 (10). Let us express
x + 3 =
2
A (5 4 ) + B
d
– x – x
dx
= A (– 4 – 2x) + B
Equating the coefficients of x and the constant terms from both sides, we get
– 2A = 1 and – 4 A + B = 3, i.e., A =
1
2
– and B = 1](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-28-2048.jpg)
![INTEGRALS 315
Therefore, 2
3
5 4
x
dx
x x
+
− −
∫ =
( )
2 2
4 2
1
2 5 4 5 4
– – x dx dx
–
x x x x
+
− − − −
∫ ∫
=
1
2
– I1
+ I2
... (1)
In I1
, put 5 – 4x – x2 = t, so that (– 4 – 2x) dx = dt.
Therefore, I1
=
( )
2
4 2
5 4
– x dx dt
t
x x
−
=
− −
∫ ∫ = 1
2 C
t +
= 2
1
2 5 4 C
– x – x + ... (2)
Now consider I2
= 2 2
5 4 9 ( 2)
dx dx
x x – x
=
− − +
∫ ∫
Put x + 2 = t, so that dx = dt.
Therefore, I2
=
1
2
2 2
sin + C
3
3
–
dt t
t
=
−
∫ [by 7.4 (5)]
=
1
2
2
sin C
3
– x +
+ ... (3)
Substituting (2) and (3) in (1), we obtain
2 1
2
3 2
5 – 4 – + sin C
3
5 4
–
x x
– x x
– x – x
+ +
= +
∫ , where 1
2
C
C C
2
–
=
EXERCISE 7.4
Integrate the functions in Exercises 1 to 23.
1.
2
6
3
1
x
x +
2. 2
1
1 4x
+
3.
( )2
1
2 1
– x +
4. 2
1
9 25
– x
5. 4
3
1 2
x
x
+
6.
2
6
1
x
x
−
7. 2
1
1
x –
x –
8.
2
6 6
x
x a
+
9.
2
2
sec
tan 4
x
x +](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-29-2048.jpg)
![318 MATHEMATICS
Example 11 Find ( 1) ( 2)
dx
x x
+ +
∫
Solution The integrand is a proper rational function. Therefore, by using the form of
partial fraction [Table 7.2 (i)], we write
1
( 1) ( 2)
x x
+ +
=
A B
1 2
x x
+
+ +
... (1)
where, real numbers A and B are to be determined suitably. This gives
1 = A (x + 2) + B (x + 1).
Equating the coefficients of x and the constant term, we get
A + B = 0
and 2A + B = 1
Solving these equations, we getA =1 and B = – 1.
Thus, the integrand is given by
1
( 1) ( 2)
x x
+ +
=
1 –1
1 2
x x
+
+ +
Therefore,
( 1) ( 2)
dx
x x
+ +
∫ =
1 2
dx dx
–
x x
+ +
∫ ∫
= log 1 log 2 C
x x
+ − + +
=
1
log C
2
x
x
+
+
+
Remark The equation (1) above is an identity, i.e. a statement true for all (permissible)
values of x. Some authors use the symbol ‘≡’ to indicate that the statement is an
identity and use the symbol ‘=’ to indicate that the statement is an equation, i.e., to
indicate that the statement is true only for certain values of x.
Example 12 Find
2
2
1
5 6
x
dx
x x
+
− +
∫
Solution Here the integrand
2
2
1
5 6
x
x – x
+
+
is not proper rational function, so we divide
x2
+ 1 by x2
– 5x + 6 and find that](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-32-2048.jpg)
![INTEGRALS 321
Therefore,
( )
2
3sin 2 cos
5 cos 4sin
–
d
– –
φ φ
φ
φ φ
∫ = 2
(3 – 2)
5 (1 ) 4
y dy
– – y – y
∫
= 2
3 2
4 4
y –
dy
y – y +
∫
=
( )2
3 2
I (say)
2
y –
y –
=
∫
Now, we write
( )2
3 2
2
y –
y –
= 2
A B
2 ( 2)
y y
+
− −
[by Table 7.2 (2)]
Therefore, 3y – 2 = A (y – 2) + B
Comparing the coefficients of y and constant term, we get A = 3 and B – 2A = – 2,
which gives A = 3 and B = 4.
Therefore, the required integral is given by
I = 2
3 4
[ + ]
2 ( 2)
dy
y – y –
∫ = 2
3 + 4
2 ( 2)
dy dy
y – y –
∫ ∫
=
1
3log 2 4 C
2
y –
y
− + +
−
=
4
3log sin 2 C
2 sin
–
φ − + +
φ
=
4
3log (2 sin ) + C
2 sin
− φ +
− φ
(since, 2 – sinφ is always positive)
Example 16 Find
2
2
1
( 2) ( 1)
x x dx
x x
+ +
+ +
∫
Solution The integrand is a proper rational function. Decompose the rational function
into partial fraction [Table 2.2(5)]. Write
2
2
1
( 1) ( 2)
x x
x x
+ +
+ +
= 2
A B + C
2 ( 1)
x
x x
+
+ +
Therefore, x2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-35-2048.jpg)
![322 MATHEMATICS
Equating the coefficients of x2
, x and of constant term of both sides, we get
A + B =1, 2B + C = 1 and A + 2C = 1. Solving these equations, we get
3 2 1
A , B and C
5 5 5
= = =
Thus, the integrand is given by
2
2
1
( 1) ( 2)
x x
x x
+ +
+ +
= 2
2 1
3 5 5
5 ( 2) 1
x
x x
+
+
+ +
= 2
3 1 2 1
5 ( 2) 5 1
x
x x
+
+
+ +
Therefore,
2
2
1
( +1) ( 2)
x x
dx
x x
+ +
+
∫ = 2 2
3 1 2 1 1
5 2 5 5
1 1
dx x
dx dx
x x x
+ +
+ + +
∫ ∫ ∫
=
2 1
3 1 1
log 2 log 1 tan C
5 5 5
–
x x x
+ + + + +
EXERCISE 7.5
Integrate the rational functions in Exercises 1 to 21.
1.
( 1) ( 2)
x
x x
+ +
2. 2
1
9
x –
3.
3 1
( 1) ( 2) ( 3)
x –
x – x – x –
4.
( 1) ( 2) ( 3)
x
x – x – x –
5. 2
2
3 2
x
x x
+ +
6.
2
1
(1 2 )
– x
x – x
7. 2
( 1) ( – 1)
x
x x
+
8. 2
( 1) ( 2)
x
x – x +
9. 3 2
3 5
1
x
x – x x
+
− +
10. 2
2 3
( 1) (2 3)
x
x – x
−
+
11. 2
5
( 1) ( 4)
x
x x
+ −
12.
3
2
1
1
x x
x
+ +
−
13. 2
2
(1 ) (1 )
x x
− +
14. 2
3 1
( 2)
x –
x +
15. 4
1
1
x −
16.
1
( 1)
n
x x +
[Hint: multiply numerator and denominator byx n – 1
and put xn
= t ]
17.
cos
(1– sin ) (2 – sin )
x
x x
[Hint : Put sin x = t]](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-36-2048.jpg)
![INTEGRALS 323
18.
2 2
2 2
( 1) ( 2)
( 3)( 4)
x x
x x
+ +
+ +
19. 2 2
2
( 1) ( 3)
x
x x
+ +
20. 4
1
( 1)
x x –
21.
1
( 1)
x
e –
[Hint : Put ex = t]
Choose the correct answer in each of the Exercises 22 and 23.
22.
( 1) ( 2)
x dx
x x
− −
∫ equals
(A)
2
( 1)
log C
2
x
x
−
+
−
(B)
2
( 2)
log C
1
x
x
−
+
−
(C)
2
1
log C
2
x
x
−
+
−
(D) log ( 1) ( 2) C
x x
− − +
23. 2
( 1)
dx
x x +
∫ equals
(A)
2
1
log log ( +1) + C
2
x x
− (B)
2
1
log log ( +1) + C
2
x x
+
(C) 2
1
log log ( +1) +C
2
x x
− + (D)
2
1
log log ( +1) + C
2
x x
+
7.6 Integration by Parts
In this section, we describe one more method of integration, that is found quite useful in
integrating products of functions.
If u and v are any two differentiable functions of a single variable x (say). Then, by
the product rule of differentiation, we have
( )
d
uv
dx
=
dv du
u v
dx dx
+
Integrating both sides, we get
uv =
dv du
u dx v dx
dx dx
+
∫ ∫
or
dv
u dx
dx
∫ =
du
uv – v dx
dx
∫ ... (1)
Let u = f (x) and
dv
dx
= g(x). Then
du
dx
= f ′(x) and v = ( )
g x dx
∫](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-37-2048.jpg)
![324 MATHEMATICS
Therefore, expression (1) can be rewritten as
( ) ( )
f x g x dx
∫ = ( ) ( ) [ ( ) ] ( )
f x g x dx – g x dx f x dx
′
∫ ∫ ∫
i.e., ( ) ( )
f x g x dx
∫ = ( ) ( ) [ ( ) ( ) ]
f x g x dx – f x g x dx dx
′
∫ ∫ ∫
If we take f as the first function and g as the second function, then this formula
may be stated as follows:
“The integral of the product of two functions = (first function) × (integral
of the second function) – Integral of [(differential coefficient of the first function)
× (integral of the second function)]”
Example 17 Find cos
x x dx
∫
Solution Put f (x) = x (first function) and g (x) = cos x (second function).
Then, integration by parts gives
cos
x x dx
∫ = cos [ ( ) cos ]
d
x x dx – x x dx dx
dx
∫ ∫ ∫
= sin sin
x x – x dx
∫ = x sin x + cos x + C
Suppose, we take f (x) = cos x and g(x) = x. Then
cos
x x dx
∫ = cos [ (cos ) ]
d
x x dx – x x dx dx
dx
∫ ∫ ∫
= ( )
2 2
cos sin
2 2
x x
x x dx
+ ∫
Thus, it shows that the integral cos
x x dx
∫ is reduced to the comparatively more
complicated integral having more power of x. Therefore, the proper choice of the first
function and the second function is significant.
Remarks
(i) It is worth mentioning that integration by parts is not applicable to product of
functions in all cases. For instance, the method does not work for sin
x x dx
∫ .
The reason is that there does not exist any function whose derivative is
x sin x.
(ii) Observe that while finding the integral of the second function, we did not add
any constant of integration. If we write the integral of the second function cos x](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-38-2048.jpg)
![INTEGRALS 325
as sin x + k, where k is any constant, then
cos
x x dx
∫ = (sin ) (sin )
x x k x k dx
+ − +
∫
= (sin ) (sin
x x k x dx k dx
+ − −
∫ ∫
= (sin ) cos C
x x k x – kx
+ − + = sin cos C
x x x
+ +
This shows that adding a constant to the integral of the second function is
superfluous so far as the final result is concerned while applying the method of
integration by parts.
(iii) Usually, if any function is a power of x or a polynomial in x, then we take it as the
first function. However, in cases where other function is inverse trigonometric
function or logarithmic function, then we take them as first function.
Example 18 Find log x dx
∫
Solution To start with, we are unable to guess a function whose derivative is log x. We
take log x as the first function and the constant function 1 as the second function. Then,
the integral of the second function is x.
Hence, (log .1)
x dx
∫ = log 1 [ (log ) 1 ]
d
x dx x dx dx
dx
−
∫ ∫ ∫
=
1
(log ) – log C
x x x dx x x – x
x
⋅ = +
∫ .
Example 19 Find
x
x e dx
∫
Solution Take first function as x and second function as ex. The integral of the second
function is ex
.
Therefore,
x
x e dx
∫ = 1
x x
x e e dx
− ⋅
∫ = xex
– ex
+ C.
Example 20 Find
1
2
sin
1
–
x x
dx
x
−
∫
Solution Let first function be sin – 1
x and second function be 2
1
x
x
−
.
First we find the integral of the second function, i.e.,
2
1
x dx
x
−
∫ .
Put t =1 – x2
. Then dt = – 2x dx](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-39-2048.jpg)
![326 MATHEMATICS
Therefore,
2
1
x dx
x
−
∫ =
1
2
dt
–
t
∫ =
2
– 1
t x
= − −
Hence,
1
2
sin
1
–
x x
dx
x
−
∫ = ( )
1 2 2
2
1
(sin ) 1 ( 1 )
1
–
x – x – x dx
x
− − −
−
∫
= 2 1
1 sin C
– x x x
−
− + + =
2 1
1 sin C
x – x x
−
− +
Alternatively, this integral can also be worked out by making substitution sin–1x = θand
then integrating by parts.
Example 21 Find sin
x
e x dx
∫
Solution Take ex
as the first function and sin x as second function. Then, integrating
by parts, we have
I sin ( cos ) cos
x x x
e x dx e – x e x dx
= = +
∫ ∫
= – ex
cos x + I1 (say) ... (1)
Taking ex
and cos x as the first and second functions, respectively, in I1, we get
I1 = sin sin
x x
e x – e x dx
∫
Substituting the value of I1
in (1), we get
I = – ex cos x + ex sin x – I or 2I = ex (sin x – cos x)
Hence, I = sin (sin cos ) + C
2
x
x e
e x dx x – x
=
∫
Alternatively, above integral can also be determined by taking sin x as the first function
and ex
the second function.
7.6.1 Integral of the type [ ( ) + ( )]
x
e f x f x dx
′
∫
We have I = [ ( ) + ( )]
x
e f x f x dx
′
∫ = ( ) + ( )
x x
e f x dx e f x dx
′
∫ ∫
= 1 1
I ( ) , where I = ( )
x x
e f x dx e f x dx
′
+ ∫ ∫ ... (1)
Taking f(x) and ex
as the first function and second function, respectively, in I1
and
integrating it by parts, we have I1 = f (x) ex
– ( ) C
x
f x e dx
′ +
∫
Substituting I1
in (1), we get
I = ( ) ( ) ( ) C
x x x
e f x f x e dx e f x dx
′ ′
− + +
∫ ∫ = ex
f (x) + C](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-40-2048.jpg)
![INTEGRALS 327
Thus, ′
∫ [ ( ) ( )]
x
e f x + f x dx = ( ) C
x
e f x +
Example 22 Find (i) 1
2
1
(tan )
1
x –
e x
x
+
+
∫ dx (ii)
2
2
( + 1)
( + 1)
x
x e
x
∫ dx
Solution
(i) We have I =
1
2
1
(tan )
1
x –
e x dx
x
+
+
∫
Consider f (x) = tan– 1
x, then f ′(x) = 2
1
1 x
+
Thus, the given integrand is of the form ex
[ f (x) + f ′(x)].
Therefore, 1
2
1
I (tan )
1
x –
e x dx
x
= +
+
∫ = ex
tan– 1
x + C
(ii) We have
2
2
( +1)
I
( +1)
x
x e
x
= ∫ dx
2
2
1 +1+1)
[ ]
( +1)
x x –
e dx
x
= ∫
2
2 2
1 2
[ ]
( +1) ( +1)
x x –
e dx
x x
= +
∫ 2
1 2
[ + ]
+1 ( +1)
x x –
e dx
x x
= ∫
Consider
1
( )
1
x
f x
x
−
=
+
, then 2
2
( )
( 1)
f x
x
′ =
+
Thus, the given integrand is of the form ex
[f (x) + f ′(x)].
Therefore,
2
2
1 1
C
1
( 1)
x x
x x
e dx e
x
x
+ −
= +
+
+
∫
EXERCISE 7.6
Integrate the functions in Exercises 1 to 22.
1. x sin x 2. x sin 3x 3. x2 ex 4. x log x
5. x log 2x 6. x2
log x 7. x sin– 1
x 8. x tan–1
x
9. x cos–1
x 10. (sin–1
x)2
11.
1
2
cos
1
x x
x
−
−
12. x sec2
x
13. tan–1x 14. x (log x)2 15. (x2 + 1) log x](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-41-2048.jpg)
![INTEGRALS 329
=
2 2 2 2 2
2 2
dx
x x a x a dx a
x a
− − − −
−
∫ ∫
=
2 2 2
2 2
I
dx
x x a a
x a
− − −
−
∫
or 2I =
2 2 2
2 2
dx
x x a a
x a
− −
−
∫
or I = ∫
2 2
x – a dx =
2
2 2 2 2
– – log + – + C
2 2
x a
x a x x a
Similarly, integrating other two integrals by parts, taking constant function 1 as the
second function, we get
(ii) ∫
2
2 2 2 2 2 2
1
+ = + + log + + + C
2 2
a
x a dx x x a x x a
(iii) ∫
2
2 2 2 2 –1
1
– = – + sin + C
2 2
a x
a x dx x a x
a
Alternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric
substitution x = a secθ in (i), x = a tanθ in (ii) and x = a sinθ in (iii) respectively.
Example 23 Find
2
2 5
x x dx
+ +
∫
Solution Note that
2
2 5
x x dx
+ +
∫ = 2
( 1) 4
x dx
+ +
∫
Put x + 1 = y, so that dx = dy. Then
2
2 5
x x dx
+ +
∫ = 2 2
2
y dy
+
∫
=
2 2
1 4
4 log 4 C
2 2
y y y y
+ + + + + [using 7.6.2 (ii)]
=
2 2
1
( 1) 2 5 2 log 1 2 5 C
2
x x x x x x
+ + + + + + + + +
Example 24 Find 2
3 2x x dx
− −
∫
Solution Note that 2 2
3 2 4 ( 1)
x x dx x dx
− − = − +
∫ ∫](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-43-2048.jpg)
![330 MATHEMATICS
Put x + 1 = y so that dx = dy.
Thus 2
3 2x x dx
− −
∫ = 2
4 y dy
−
∫
=
2 1
1 4
4 sin C
2 2 2
– y
y y
− + + [using 7.6.2 (iii)]
=
2 1
1 1
( 1) 3 2 2 sin C
2 2
– x
x x x
+
+ − − + +
EXERCISE 7.7
Integrate the functions in Exercises 1 to 9.
1. 2
4 x
− 2. 2
1 4x
− 3. 2
4 6
x x
+ +
4. 2
4 1
x x
+ + 5. 2
1 4x x
− − 6. 2
4 5
x x
+ −
7. 2
1 3x x
+ − 8. 2
3
x x
+ 9.
2
1
9
x
+
Choose the correct answer in Exercises 10 to 11.
10. 2
1 x dx
+
∫ is equal to
(A) ( )
2 2
1
1 log 1 C
2 2
x
x x x
+ + + + +
(B)
3
2 2
2
(1 ) C
3
x
+ + (C)
3
2 2
2
(1 ) C
3
x x
+ +
(D)
2
2 2 2
1
1 log 1 C
2 2
x
x x x x
+ + + + +
11. 2
8 7
x x dx
− +
∫ is equal to
(A)
2 2
1
( 4) 8 7 9log 4 8 7 C
2
x x x x x x
− − + + − + − + +
(B)
2 2
1
( 4) 8 7 9log 4 8 7 C
2
x x x x x x
+ − + + + + − + +
(C)
2 2
1
( 4) 8 7 3 2log 4 8 7 C
2
x x x x x x
− − + − − + − + +
(D)
2 2
1 9
( 4) 8 7 log 4 8 7 C
2 2
x x x x x x
− − + − − + − + +](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-44-2048.jpg)
![INTEGRALS 331
7.7 Definite Integral
In the previous sections, we have studied about the indefinite integrals and discussed
few methods of finding them including integrals of some special functions. In this
section, we shall study what is called definite integral of a function. The definite integral
has a unique value.A definite integral is denoted by ( )
b
a
f x dx
∫ , wherea is called the
lower limit of the integral and b is called the upper limit of the integral. The definite
integral is introduced either as the limit of a sum or if it has an anti derivative F in the
interval [a, b], then its value is the difference between the values of F at the end
points, i.e., F(b) – F(a). Here, we shall consider these two cases separately as discussed
below:
7.7.1 Definite integral as the limit of a sum
Let f be a continuous function defined on close interval [a, b]. Assume that all the
values taken by the function are non negative, so the graph of the function is a curve
above the x-axis.
The definite integral ( )
b
a
f x dx
∫ is the area bounded by the curve y = f (x), the
ordinates x = a, x = b and the x-axis.To evaluate this area, consider the region PRSQP
between this curve, x-axis and the ordinates x = a and x = b (Fig 7.2).
Divide the interval [a, b] into n equal subintervals denoted by [x0
, x1
], [x1
, x2
] ,...,
[xr – 1
, xr
], ..., [xn – 1
, xn
], where x0
= a, x1
= a + h, x2
= a + 2h, ... , xr
= a + rh and
xn
= b = a + nh or .
b a
n
h
−
= We note that as n → ∞, h → 0.
Fig 7.2
O
Y
X
X'
Y'
Q
P
C
M
D
L
S
A B R
a= x0 x1 x2 xr-1 xr x =b
n
y
f x
=
( )](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-45-2048.jpg)
![332 MATHEMATICS
The region PRSQP under consideration is the sum of n subregions, where each
subregion is defined on subintervals [xr – 1
, xr
], r = 1, 2, 3, …, n.
From Fig 7.2, we have
area of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle
(ABDM) ... (1)
Evidently as xr – xr–1 → 0, i.e., h → 0 all the three areas shown in (1) become
nearly equal to each other. Now we form the following sums.
sn
= h [f(x0
) + … + f (xn - 1
)] =
1
0
( )
n
r
r
h f x
−
=
∑ ... (2)
and Sn
= 1 2
1
[ ( ) ( ) ( )] ( )
n
n r
r
h f x f x f x h f x
=
+ +…+ = ∑ ... (3)
Here, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles
raised over subintervals [xr–1
, xr
] for r = 1, 2, 3, …, n, respectively.
In view of the inequality (1) for an arbitrary subinterval [xr–1
, xr
], we have
sn
< area of the region PRSQP < Sn
... (4)
As n → ∞ strips become narrower and narrower, it is assumed that the limiting
values of (2) and (3) are the same in both cases and the common limiting value is the
required area under the curve.
Symbolically, we write
limSn
n→∞
= lim n
n
s
→∞
= area of the region PRSQP = ( )
b
a
f x dx
∫ ... (5)
It follows that this area is also the limiting value of any area which is between that
of the rectangles below the curve and that of the rectangles above the curve. For
the sake of convenience, we shall take rectangles with height equal to that of the
curve at the left hand edge of each subinterval. Thus, we rewrite (5) as
( )
b
a
f x dx
∫ =
0
lim [ ( ) ( ) ... ( ( – 1) ]
h
h f a f a h f a n h
→
+ + + + +
or ( )
b
a
f x dx
∫ =
1
( – ) lim [ ( ) ( ) ... ( ( –1) ]
n
b a f a f a h f a n h
n
→∞
+ + + + + ... (6)
where h =
–
0
b a
as n
n
→ → ∞
The above expression (6) is known as the definition of definite integral as the limit
of sum.
Remark The value of the definite integral of a function over any particular interval
depends on the function and the interval, but not on the variable of integration that we](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-46-2048.jpg)
![INTEGRALS 333
choose to represent the independent variable. If the independent variable is denoted by
t or u instead ofx, we simply write the integral as ( )
b
a
f t dt
∫ or ( )
b
a
f u du
∫ instead of
( )
b
a
f x dx
∫ . Hence, the variable of integration is called a dummy variable.
Example 25 Find
2
2
0
( 1)
x dx
+
∫ as the limit of a sum.
Solution By definition
( )
b
a
f x dx
∫ =
1
( – ) lim [ ( ) ( ) ... ( ( –1) ],
n
b a f a f a h f a n h
n
→∞
+ + + + +
where, h =
–
b a
n
In this example, a = 0, b = 2, f (x) = x2
+ 1,
2 – 0 2
h
n n
= =
Therefore,
2 2
0
( 1)
x dx
+
∫ =
1 2 4 2 ( –1)
2 lim [ (0) ( ) ( ) ... ( )]
n
n
f f f f
n n n n
→∞
+ + + +
=
2 2 2
2 2 2
1 2 4 (2 – 2)
2 lim [1 ( 1) ( 1) ... 1 ]
n
n
n n n n
→∞
+ + + + + + +
=
2 2 2
-
1 1
2 lim [(1 1 ... 1) (2 4 ... (2 – 2) ]
2
→∞
+ + + + + + +
14
4
244
3
n
n terms
n
n n
=
2
2 2 2
1 2
2 lim [ (1 2 ... ( –1) ]
n
n n
n n2
→∞
+ + + +
=
1 4 ( 1) (2 –1)
2 lim [ ]
6
n
n n n
n
n n2
→∞
−
+
=
1 2 ( 1) (2 –1)
2 lim [ ]
3
n
n n
n
n n
→∞
−
+
=
2 1 1
2 lim [1 (1 ) (2 – )]
3
n n n
→∞
+ − =
4
2 [1 ]
3
+ =
14
3](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-47-2048.jpg)
![334 MATHEMATICS
Example 26 Evaluate
2
0
x
e dx
∫ as the limit of a sum.
Solution By definition
2
0
x
e dx
∫ =
2 4 2 – 2
0
1
(2 – 0) lim ...
n
n n n
n
e e e e
n
→∞
+ + + +
Using the sum to n terms of a G.P., where a = 1,
2
n
r e
= , we have
2
0
x
e dx
∫ =
2
2
1 –1
2 lim [ ]
1
n
n
n
n
e
n
e
→∞
−
=
2
2
1 –1
2 lim
–1
n
n
e
n
e
→∞
=
2
2
2 ( –1)
–1
lim 2
2
n
n
e
e
n
→∞
⋅
= e2
– 1 [using
0
( 1)
lim 1
h
h
e
h
→
−
= ]
EXERCISE 7.8
Evaluate the following definite integrals as limit of sums.
1.
b
a
x dx
∫ 2.
5
0
( 1)
x dx
+
∫ 3.
3 2
2
x dx
∫
4.
4 2
1
( )
x x dx
−
∫ 5.
1
1
x
e dx
−
∫ 6.
4 2
0
( )
x
x e dx
+
∫
7.8 Fundamental Theorem of Calculus
7.8.1 Area function
We have defined ( )
b
a
f x dx
∫ as the area of
the region bounded by the curve y = f(x),
the ordinates x = a and x = b and x-axis. Let x
be a given point in [a, b]. Then ( )
x
a
f x dx
∫
represents the area of the light shaded region Fig 7.3](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-48-2048.jpg)
![INTEGRALS 335
in Fig 7.3 [Here it is assumed that f (x) > 0 for x ∈ [a, b], the assertion made below is
equally true for other functions as well]. The area of this shaded region depends upon
the value of x.
In other words, the area of this shaded region is a function of x. We denote this
function of x by A(x). We call the function A(x) as Area function and is given by
A (x) = ∫ ( )
x
a
f x dx ... (1)
Based on this definition, the two basic fundamental theorems have been given.
However, we only state them as their proofs are beyond the scope of this text book.
7.8.2 First fundamental theorem of integral calculus
Theorem 1 Let f be a continuous function on the closed interval [a, b] and letA (x) be
the area function. Then A′
′
′
′
′(x) = f (x), for all x ∈
∈
∈
∈
∈ [a, b].
7.8.3 Second fundamental theorem of integral calculus
We state below an important theorem which enables us to evaluate definite integrals
by making use of anti derivative.
Theorem 2 Let f be continuous function defined on the closed interval [a, b] and F be
an anti derivative of f. Then ∫ ( )
b
a
f x dx = [F( )] =
b
a
x F (b) – F(a).
Remarks
(i) In words, the Theorem 2 tells us that ( )
b
a
f x dx
∫ = (value of the anti derivative F
of f at the upper limit b – value of the same anti derivative at the lower limit a).
(ii) This theorem is very useful, because it gives us a method of calculating the
definite integral more easily, without calculating the limit of a sum.
(iii) The crucial operation in evaluating a definite integral is that of finding a function
whose derivative is equal to the integrand. This strengthens the relationship
between differentiation and integration.
(iv) In ( )
b
a
f x dx
∫ , the functionf needs to be well defined and continuous in [a, b].
For instance, the consideration of definite integral
1
3 2 2
2
( –1)
x x dx
−
∫ is erroneous
since the function f expressed by f(x) =
1
2 2
( –1)
x x is not defined in a portion
– 1 < x < 1 of the closed interval [– 2, 3].](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-49-2048.jpg)
![336 MATHEMATICS
Steps for calculating ( )
b
a
f x dx
∫ .
(i) Find the indefinite integral ( )
f x dx
∫ . Let this be F(x). There is no need to keep
integration constant C because if we consider F(x) + C instead of F(x), we get
( ) [F ( ) C] [F( ) C]– [F( ) C] F( ) – F( )
b b
a
a
f x dx x b a b a
= + = + + =
∫ .
Thus, the arbitrary constant disappears in evaluating the value of the definite
integral.
(ii) Evaluate F(b) – F(a) = [F ( )]b
a
x , which is the value of ( )
b
a
f x dx
∫ .
We now consider some examples
Example 27 Evaluate the following integrals:
(i)
3
2
2
x dx
∫ (ii)
9
3
4
2
2
(30– )
x
dx
x
∫
(iii)
2
1 ( 1) ( 2)
x dx
x x
+ +
∫ (iv) 3
4
0
sin 2 cos 2
t t dt
π
∫
Solution
(i) Let
3 2
2
I x dx
= ∫ . Since
3
2
F ( )
3
x
x dx x
= =
∫ ,
Therefore, by the second fundamental theorem, we get
I =
27 8 19
F (3) – F (2) –
3 3 3
= =
(ii) Let
9
3
4
2
2
I
(30 – )
x
dx
x
= ∫ . We first find the anti derivative of the integrand.
Put
3
2
3
30 – . Then –
2
x t x dx dt
= = or
2
–
3
x dx dt
=
Thus, 3 2
2
2
2
–
3
(30 – )
x dt
dx
t
x
=
∫ ∫ =
2 1
3 t
= 3
2
2 1
F ( )
3
(30 – )
x
x
=
](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-50-2048.jpg)
![INTEGRALS 337
Therefore, by the second fundamental theorem of calculus, we have
I =
9
3
2
4
2 1
F(9) – F(4)
3
(30 – )
x
=
=
2 1 1
3 (30 – 27) 30 –8
−
=
2 1 1 19
3 3 22 99
− =
(iii) Let
2
1
I
( 1) ( 2)
x dx
x x
=
+ +
∫
Using partial fraction, we get
–1 2
( 1) ( 2) 1 2
x
x x x x
= +
+ + + +
So
( 1) ( 2)
x dx
x x
+ +
∫ = – log 1 2log 2 F( )
x x x
+ + + =
Therefore, by the second fundamental theorem of calculus, we have
I = F(2) – F(1) = [– log 3 + 2 log 4] – [– log 2 + 2 log 3]
= – 3 log 3 + log 2 + 2 log 4 =
32
log
27
(iv) Let 3
4
0
I sin 2 cos2
t t dt
π
= ∫ . Consider
3
sin 2 cos2
t t dt
∫
Put sin 2t = u so that 2 cos 2t dt = du or cos 2t dt =
1
2
du
So
3
sin 2 cos2
t t dt
∫ =
3
1
2
u du
∫
=
4 4
1 1
[ ] sin 2 F ( )say
8 8
u t t
= =
Therefore, by the second fundamental theorem of integral calculus
I = 4 4
1 1
F ( ) – F (0) [sin –sin 0]
4 8 2 8
π π
= =](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-51-2048.jpg)
![INTEGRALS 341
7.10 Some Properties of Definite Integrals
We list below some important properties of definite integrals.These will be useful in
evaluating the definite integrals more easily.
P0
: ( ) ( )
b b
a a
f x dx f t dt
=
∫ ∫
P1
: ( ) – ( )
b a
a b
f x dx f x dx
=
∫ ∫ . In particular, ( ) 0
a
a
f x dx =
∫
P2
: ( ) ( ) ( )
b c b
a a c
f x dx f x dx f x dx
= +
∫ ∫ ∫
P3
: ( ) ( )
b b
a a
f x dx f a b x dx
= + −
∫ ∫
P4 :
0 0
( ) ( )
a a
f x dx f a x dx
= −
∫ ∫
(Note that P4
is a particular case of P3
)
P5 :
2
0 0 0
( ) ( ) (2 )
a a a
f x dx f x dx f a x dx
= + −
∫ ∫ ∫
P6
:
2
0 0
( ) 2 ( ) , if (2 ) ( )
a a
f x dx f x dx f a x f x
= − =
∫ ∫ and
0 if f (2a – x) = – f (x)
P7
: (i)
0
( ) 2 ( )
a a
a
f x dx f x dx
−
=
∫ ∫ , if f is an even function, i.e., if f (–x) = f (x).
(ii) ( ) 0
a
a
f x dx
−
=
∫ , if f is an odd function, i.e., if f (– x) = – f (x).
We give the proofs of these properties one by one.
Proof of P0
It follows directly by making the substitution x = t.
Proof of P1
Let F be anti derivative of f. Then, by the second fundamental theorem of
calculus, we have ( ) F ( ) – F ( ) –[F ( ) F ( )] ( )
b a
a b
f x dx b a a b f x dx
= = − = −
∫ ∫
Here, we observe that, if a = b, then ( ) 0
a
a
f x dx =
∫ .
Proof of P2 Let F be anti derivative of f. Then
( )
b
a
f x dx
∫ = F(b) – F(a) ... (1)
( )
c
a
f x dx
∫ = F(c) – F(a) ... (2)
and ( )
b
c
f x dx
∫ = F(b) – F(c) ... (3)](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-55-2048.jpg)
![INTEGRALS 343
Therefore ( )
a
a
f x dx
−
∫ =
0
0
– (– ) ( )
a
a
f t dt f x dx
+
∫ ∫
= 0 0
(– ) ( )
a a
f x dx f x dx
+
∫ ∫ (by P0
) ... (1)
(i) Now, if f is an even function, then f (–x) = f (x) and so (1) becomes
0 0 0
( ) ( ) ( ) 2 ( )
a a a a
a
f x dx f x dx f x dx f x dx
−
= + =
∫ ∫ ∫ ∫
(ii) If f is an odd function, then f (–x) = – f(x) and so (1) becomes
0 0
( ) ( ) ( ) 0
a a a
a
f x dx f x dx f x dx
−
= − + =
∫ ∫ ∫
Example 30 Evaluate
2
3
1
–
x x dx
−
∫
Solution We note that x3 – x ≥ 0 on [– 1, 0] and x3 – x ≤ 0 on [0, 1] and that
x3 – x ≥ 0 on [1, 2]. So by P2
we write
2 3
1
–
x x dx
−
∫ =
0 1 2
3 3 3
1 0 1
( – ) – ( – ) ( – )
x x dx x x dx x x dx
−
+ +
∫ ∫ ∫
=
0 1 2
3 3 3
1 0 1
( – ) ( – ) ( – )
x x dx x x dx x x dx
−
+ +
∫ ∫ ∫
=
0 1 2
4 2 2 4 4 2
–1 0 1
– – –
4 2 2 4 4 2
x x x x x x
+ +
= ( )
1 1 1 1 1 1
– – – 4 – 2 – –
4 2 2 4 4 2
+ +
=
1 1 1 1 1 1
– 2
4 2 2 4 4 2
+ + − + − + =
3 3 11
2
2 4 4
− + =
Example 31 Evaluate 2
4
–
4
sin x dx
π
π
∫
Solution We observe that sin2
x is an even function. Therefore, by P7 (i), we get
2
4
–
4
sin x dx
π
π
∫ =
2
4
0
2 sin x dx
π
∫](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-57-2048.jpg)
![INTEGRALS 345
Example 34 Evaluate
4
2
4 4
0
sin
sin cos
x
dx
x x
π
+
∫
Solution Let I =
4
2
4 4
0
sin
sin cos
x
dx
x x
π
+
∫ ... (1)
Then, by P4
I =
4
2
0 4 4
sin ( )
2
sin ( ) cos ( )
2 2
x
dx
x x
π
π
−
π π
− + −
∫ =
4
2
4 4
0
cos
cos sin
x
dx
x x
π
+
∫ ... (2)
Adding (1) and (2), we get
2I =
4 4
2
2 2
4 4
0 0 0
sin cos
[ ]
2
sin cos
x x
dx dx x
x x
π
π π
+ π
= = =
+
∫ ∫
Hence I =
4
π
Example 35 Evaluate
3
6
1 tan
dx
x
π
π
+
∫
Solution Let I = 3 3
6 6
cos
1 tan cos sin
x dx
dx
x x x
π π
π π
=
+ +
∫ ∫ ... (1)
Then, by P3 I =
3
6
cos
3 6
cos sin
3 6 3 6
x dx
x x
π
π
π π
+ −
π π π π
+ − + + −
∫
=
3
6
sin
sin cos
x
dx
x x
π
π
+
∫ ... (2)
Adding (1) and (2), we get
2I = [ ]
3 3
6 6
3 6 6
dx x
π π
π π
π π π
= = − =
∫ . Hence I
12
π
=](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-59-2048.jpg)
![352 MATHEMATICS
or I = 2
2 2 2 2 2 2 2 2
0 0
2
2 2
cos sin cos sin
dx dx
a x b x a x b x
π
π
π π
= ⋅
+ +
∫ ∫ (using P6
)
=
2
4
2 2 2 2 2 2 2 2
0
4
cos sin cos sin
π
π
π
π +
+ +
∫ ∫
dx dx
a x b x a x b x
=
2 2
2
4
2 2 2 2 2 2
0
4
sec cosec
tan cot
π
π
π
π +
+ +
∫ ∫
xdx xdx
a b x a x b
= ( )
0
1
2 2 2 2 2 2
0 1
tan t cot
π − = =
+ +
∫ ∫
dt du
put x and x u
a b t a u b
=
1 0
–1 –1
0 1
tan – tan
π π
bt au
ab a ab b
=
–1 –1
tan tan
π
+
b a
ab a b
=
2
2
π
ab
Miscellaneous Exercise on Chapter 7
Integrate the functions in Exercises 1 to 24.
1. 3
1
x x
−
2.
1
x a x b
+ + +
3. 2
1
x ax x
−
[Hint:Putx =
a
t
]
4. 3
2 4 4
1
( 1)
x x +
5. 1
1
3
2
1
x x
+
[Hint: 1
1 1 1
3
2 3 6
1 1
1
x x x x
=
+ +
, put x = t6
]
6. 2
5
( 1) ( 9)
x
x x
+ +
7.
sin
sin ( )
x
x a
−
8.
5 log 4 log
3 log 2 log
x x
x x
e e
e e
−
−
9. 2
cos
4 sin
x
x
−
10.
8 8
2 2
sin cos
1 2sin cos
x
x x
−
−
11.
1
cos ( ) cos ( )
x a x b
+ +
12.
3
8
1
x
x
−
13.
(1 ) (2 )
x
x x
e
e e
+ +
14. 2 2
1
( 1) ( 4)
x x
+ +
15. cos3x elog sinx 16. e3 logx (x4 + 1)– 1 17. f ′ (ax + b) [f (ax + b)]n
18. 3
1
sin sin ( )
x x + α
19.
1 1
1 1
sin cos
sin cos
x x
x x
− −
− −
−
+
, x ∈ [0, 1]](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-66-2048.jpg)
![INTEGRALS 353
20.
1
1
x
x
−
+
21.
2 sin 2
1 cos2
x
x
e
x
+
+
22.
2
2
1
( 1) ( 2)
x x
x x
+ +
+ +
23.
– 1 1
tan
1
x
x
−
+
24.
2 2
4
1 log ( 1) 2 log
x x x
x
+ + −
Evaluate the definite integrals in Exercises 25 to 33.
25.
2
1 sin
1 cos
π
π
−
−
∫
x x
e dx
x
26. 4
4 4
0
sin cos
cos sin
x x
dx
x x
π
+
∫ 27.
2
2
2 2
0
cos
cos 4 sin
x dx
x x
π
+
∫
28. 3
6
sin cos
sin 2
x x
dx
x
π
π
+
∫ 29.
1
0
1
dx
x x
+ −
∫ 30. 4
0
sin cos
9 16 sin 2
x x
dx
x
π
+
+
∫
31. 1
2
0
sin 2 tan (sin )
x x dx
π
−
∫ 32. 0
tan
sec tan
x x
dx
x x
π
+
∫
33.
4
1
[ 1| | 2| | 3|]
x x x dx
− + − + −
∫
Prove the following (Exercises 34 to 39)
34.
3
2
1
2 2
log
3 3
( 1)
dx
x x
= +
+
∫ 35.
1
0
1
x
x e dx =
∫
36.
1 17 4
1
cos 0
x x dx
−
=
∫ 37. 3
2
0
2
sin
3
x dx
π
=
∫
38. 3
4
0
2 tan 1 log2
x dx
π
= −
∫ 39.
1 1
0
sin 1
2
x dx
− π
= −
∫
40. Evaluate
1 2 3
0
x
e dx
−
∫ as a limit of a sum.
Choose the correct answers in Exercises 41 to 44.
41. x x
dx
e e−
+
∫ is equal to
(A) tan–1 (ex) + C (B) tan–1 (e–x) + C
(C) log (ex
– e–x
) + C (D) log (ex
+ e–x
) + C
42. 2
cos2
(sin cos )
x
dx
x x
+
∫ is equal to
(A)
–1
C
sin cos
x x
+
+
(B) log |sin cos | C
x x
+ +
(C) log |sin cos | C
x x
− + (D) 2
1
(sin cos )
x x
+](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-67-2048.jpg)
![354 MATHEMATICS
43. If f (a + b – x) = f (x), then ( )
b
a
x f x dx
∫ is equal to
(A) ( )
2
b
a
a b
f b x dx
+
−
∫ (B) ( )
2
b
a
a b
f b x dx
+
+
∫
(C) ( )
2
b
a
b a
f x dx
−
∫ (D) ( )
2
b
a
a b
f x dx
+
∫
44. The value of
1
1
2
0
2 1
tan
1
x
dx
x x
− −
+ −
∫ is
(A) 1 (B) 0 (C) –1 (D)
4
π
Summary
® Integration is the inverse process of differentiation. In the differential calculus,
we are given a function and we have to find the derivative or differential of
this function, but in the integral calculus, we are to find a function whose
differential is given. Thus, integration is a process which is the inverse of
differentiation.
Let F( ) ( )
d
x f x
dx
= . Then we write ( ) F ( ) C
f x dx x
= +
∫ . These integrals
are called indefinite integrals or general integrals, C is called constant of
integration.All these integrals differ by a constant.
® From the geometric point of view, an indefinite integral is collection of family
of curves, each of which is obtained by translating one of the curves parallel
to itself upwards or downwards along the y-axis.
® Some properties of indefinite integrals are as follows:
1. [ ( ) ( )] ( ) ( )
f x g x dx f x dx g x dx
+ = +
∫ ∫ ∫
2. For any real number k, ( ) ( )
k f x dx k f x dx
=
∫ ∫
More generally, if f1, f2, f3, ... , fn are functions and k1, k2, ... ,kn are real
numbers. Then
1 1 2 2
[ ( ) ( ) ... ( )]
n n
k f x k f x k f x dx
+ + +
∫
= 1 1 2 2
( ) ( ) ... ( )
n n
k f x dx k f x dx k f x dx
+ + +
∫ ∫ ∫](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-68-2048.jpg)
![INTEGRALS 357
(iv) 2 2
2 2
log C
dx
x x a
x a
= + − +
−
∫ (v)
1
2 2
sin C
dx x
a
a x
−
= +
−
∫
(vi)
2 2
2 2
log | | C
dx
x x a
x a
= + + +
+
∫
® Integration by parts
For given functions f1
and f2
, we have
1 2 1 2 1 2
( ) ( ) ( ) ( ) ( ) ( )
d
f x f x dx f x f x dx f x f x dx dx
dx
⋅ = − ⋅
∫ ∫ ∫ ∫ , i.e., the
integral of the product of two functions = first function × integral of the
second function – integral of {differential coefficient of the first function ×
integral of the second function}. Care must be taken in choosing the first
function and the second function. Obviously, we must take that function as
the second function whose integral is well known to us.
® [ ( ) ( )] ( ) C
x x
e f x f x dx e f x dx
′
+ = +
∫ ∫
® Some special types of integrals
(i)
2
2 2 2 2 2 2
log C
2 2
x a
x a dx x a x x a
− = − − + − +
∫
(ii)
2
2 2 2 2 2 2
log C
2 2
x a
x a dx x a x x a
+ = + + + + +
∫
(iii)
2
2 2 2 2 1
sin C
2 2
x a x
a x dx a x
a
−
− = − + +
∫
(iv) Integrals of the types 2 2
or
dx dx
ax bx c ax bx c
+ + + +
∫ ∫ can be
transformed into standard form by expressing
ax2
+ bx + c =
2 2
2
2
2 4
b c b c b
a x x a x
a a a a a
+ + = + + −
(v) Integrals of the types 2 2
or
px q dx px q dx
ax bx c ax bx c
+ +
+ + + +
∫ ∫ can be](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-71-2048.jpg)
![358 MATHEMATICS
transformed into standard form by expressing
2
A ( ) B A (2 ) B
d
px q ax bx c ax b
dx
+ = + + + = + + , where Aand B are
determined by comparing coefficients on both sides.
® We have defined ( )
b
a
f x dx
∫ as the area of the region bounded by the curve
y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b. Let x be a
given point in [a, b]. Then ( )
x
a
f x dx
∫ represents the Area function A (x).
This concept of area function leads to the Fundamental Theorems of Integral
Calculus.
® First fundamental theorem of integral calculus
Let the area function be defined by A(x) = ( )
x
a
f x dx
∫ for all x ≥ a, where
the function f is assumed to be continuous on [a, b]. ThenA′(x) = f (x) for all
x ∈ [a, b].
® Second fundamental theorem of integral calculus
Let f be a continuous function of x defined on the closed interval [a, b] and
let F be another function such that F( ) ( )
d
x f x
dx
= for all x in the domain of
f, then [ ]
( ) F( ) C F ( ) F ( )
b b
a
a
f x dx x b a
= + = −
∫ .
This is called the definite integral of f over the range [a, b], where a and b
are called the limits of integration, a being the lower limit and b the
upper limit.
—v
v
v
v
v—](https://image.slidesharecdn.com/chapter1-210329153903/75/Integration-material-72-2048.jpg)
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Integration material
- 1. INTEGRALS 287 vJust asa mountaineer climbs a mountain – because it is there, so a good mathematics student studies new material because it is there. — JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred on the concept of the derivative. The original motivation for the derivative was the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. If a function f is differentiable in an interval I, i.e., its derivative f ′exists at each point of I, then a natural question arises that given f ′at each point of I, can we determine the function? The functions that could possibly have given function as a derivative are called anti derivatives (or primitive) of the function. Further, the formula that gives all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. Such type of problems arise in many practical situations. For instance, if we know the instantaneous velocity of an object at any instant, then there arises a natural question, i.e., can we determine the position of the object at any instant? There are several such practical and theoretical situations where the process of integration is involved. The development of integral calculus arises out of the efforts of solving the problems of the following types: (a) the problem of finding a function whenever its derivative is given, (b) the problem of finding the area bounded by the graph of a function under certain conditions. These two problems lead to the two forms of the integrals, e.g., indefinite and definite integrals, which together constitute the Integral Calculus. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716)
- 2. 288 MATHEMATICS There isa connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical toolforscienceandengineering.Thedefiniteintegralisalsousedtosolvemanyinteresting problems from various disciplines like economics, finance and probability. In this Chapter, we shall confine ourselves to the study of indefinite and definite integrals and their elementary properties including some techniques of integration. 7.2 Integration as an Inverse Process of Differentiation Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation. Let us consider the following examples: We know that (sin ) d x dx = cos x ... (1) 3 ( ) 3 d x dx = x2 ... (2) and ( ) x d e dx = ex ... (3) We observe that in (1), the function cos x is the derived function of sin x. We say that sin xis an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), 3 3 x and ex are the anti derivatives (or integrals) of x2 and ex , respectively. Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows : (sin + C) cos = d x x dx , 3 2 ( + C) 3 = d x x dx and ( + C) = x x d e e dx Thus, anti derivatives (or integrals) of the above cited functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. More generally, if there is a function F such that F ( ) = ( ) d x f x dx , ∀ x ∈ I (interval), then for any arbitrary real number C, (also called constant of integration) [ ] F ( ) + C d x dx = f (x), x ∈ I
- 3. INTEGRALS 289 Thus, {F+ C, C ∈ R} denotes a family of anti derivatives of f. Remark Functions with same derivatives differ by a constant.To show this, let g and h be two functions having the same derivatives on an interval I. Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I Then df dx = f′ = g′ – h′ giving f′ (x) = g′(x) – h′ (x) ∀ x ∈ I or f′ (x) = 0, ∀x ∈ I by hypothesis, i.e., the rate of change of f with respect to x is zero on I and hence f is constant. In view of the above remark, it is justified to infer that the family {F + C, C ∈ R} provides all possible anti derivatives of f. We introduce a new symbol, namely, ( ) f x dx ∫ which will represent the entire class of anti derivatives read as the indefinite integral of f with respect to x. Symbolically, we write ( ) = F ( ) + C f x dx x ∫ . Notation Given that ( ) dy f x dx = , we write y = ( ) f x dx ∫ . For the sake of convenience, we mention below the following symbols/terms/phrases with their meanings as given in the Table (7.1). Table 7.1 Symbols/Terms/Phrases Meaning ( ) f x dx ∫ Integral of f with respect to x f (x) in ( ) f x dx ∫ Integrand x in ( ) f x dx ∫ Variable of integration Integrate Find the integral An integral of f A function F such that F′(x) = f (x) Integration The process of finding the integral Constant of Integration Any real number C, considered as constant function
- 4. 290 MATHEMATICS We alreadyknow the formulae for the derivatives of many important functions. From these formulae, we can write down immediately the corresponding formulae (referred to as standard formulae) for the integrals of these functions, as listed below which will be used to find integrals of other functions. Derivatives Integrals (Anti derivatives) (i) 1 1 n n d x x dx n + = + ; 1 C 1 n n x x dx n + = + + ∫ , n ≠ –1 Particularly, we note that ( ) 1 d x dx = ; C dx x = + ∫ (ii) ( ) sin cos d x x dx = ; cos sin C x dx x = + ∫ (iii) ( ) – cos sin d x x dx = ; sin cos C x dx – x = + ∫ (iv) ( ) 2 tan sec d x x dx = ; 2 sec tan C x dx x = + ∫ (v) ( ) 2 – cot cosec d x x dx = ; 2 cosec cot C x dx – x = + ∫ (vi) ( ) sec sec tan d x x x dx = ; sec tan sec C x x dx x = + ∫ (vii) ( ) – cosec cosec cot d x x x dx = ; cosec cot – cosec C x x dx x = + ∫ (viii) ( ) – 1 2 1 sin 1 d x dx – x = ; – 1 2 sin C 1 dx x – x = + ∫ (ix) ( ) – 1 2 1 – cos 1 d x dx – x = ; – 1 2 cos C 1 dx – x – x = + ∫ (x) ( ) – 1 2 1 tan 1 d x dx x = + ; – 1 2 tan C 1 dx x x = + + ∫ (xi) ( ) – 1 2 1 – cot 1 d x dx x = + ; – 1 2 cot C 1 dx – x x = + + ∫
- 5. INTEGRALS 291 (xii) () – 1 2 1 sec 1 d x dx x x – = ; – 1 2 sec C 1 dx x x x – = + ∫ (xiii) ( ) – 1 2 1 – cosec 1 d x dx x x – = ; – 1 2 cosec C 1 dx – x x x – = + ∫ (xiv) ( ) x x d e e dx = ; C x x e dx e = + ∫ (xv) ( ) 1 log| | d x dx x = ; 1 log| | C dx x x = + ∫ (xvi) x x d a a dx log a = ; C x x a a dx log a = + ∫ ANote In practice, we normally do not mention the interval over which the various functions are defined. However, in any specific problem one has to keep it in mind. 7.2.1 Geometrical interpretation of indefinite integral Let f (x) = 2x. Then 2 ( ) C f x dx x = + ∫ . For different values of C, we get different integrals. But these integrals are very similar geometrically. Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis. Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin. The curve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along y-axis in positive direction. For C = – 1, y = x2 – 1 is obtained by shifting the parabola y = x2 one unit along y-axis in the negative direction. Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig 7.1. Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x2 , y = x2 + 1, y = x2 + 2, y = x2 – 1, y = x2 – 2 at P0, P1, P2, P–1 , P–2 etc., respectively, then dy dx at these points equals 2a. This indicates that the tangents to the curves at these points are parallel. Thus, 2 C 2 C F ( ) x dx x x = + = ∫ (say), implies that
- 6. 292 MATHEMATICS the tangentsto all the curves y = FC (x), C ∈ R, at the points of intersection of the curves by the line x = a, (a ∈ R), are parallel. Further, the following equation (statement) ( ) F ( ) C (say) f x dx x y = + = ∫ , represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral. 7.2.2 Some properties of indefinite integral In this sub section, we shall derive some properties of indefinite integrals. (I) The process of differentiation and integration are inverses of each other in the sense of the following results : ( ) d f x dx dx ∫ = f (x) and ( ) f x dx ′ ∫ = f (x) + C, where C is any arbitrary constant. Fig 7.1
- 7. INTEGRALS 293 Proof LetF be any anti derivative of f, i.e., F( ) d x dx = f (x) Then ( ) f x dx ∫ = F(x) + C Therefore ( ) d f x dx dx ∫ = ( ) F ( ) + C d x dx = F ( ) = ( ) d x f x dx Similarly, we note that f ′(x) = ( ) d f x dx and hence ( ) f x dx ′ ∫ = f (x) + C where C is arbitrary constant called constant of integration. (II) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. Proof Let f and g be two functions such that ( ) d f x dx dx ∫ = ( ) d g x dx dx ∫ or ( ) ( ) d f x dx – g x dx dx ∫ ∫ = 0 Hence ( ) ( ) f x dx – g x dx ∫ ∫ = C, where C is any real number (Why?) or ( ) f x dx ∫ = ( ) C g x dx + ∫ So the families of curves { } 1 1 ( ) C ,C R f x dx + ∈ ∫ and { } 2 2 ( ) C , C R g x dx + ∈ ∫ are identical. Hence, in this sense, ( ) and ( ) f x dx g x dx ∫ ∫ are equivalent.
- 8. 294 MATHEMATICS A NoteThe equivalence of the families { } 1 1 ( ) + C ,C f x dx ∈ ∫ R and { } 2 2 ( ) + C ,C g x dx ∈ ∫ R is customarily expressed by writing ( ) = ( ) f x dx g x dx ∫ ∫ , without mentioning the parameter. (III) [ ] ( ) + ( ) ( ) + ( ) f x g x dx f x dx g x dx = ∫ ∫ ∫ Proof By Property (I), we have [ ( ) + ( )] d f x g x dx dx ∫ = f (x) + g(x) ... (1) On the otherhand, we find that ( ) + ( ) d f x dx g x dx dx ∫ ∫ = ( ) + ( ) d d f x dx g x dx dx dx ∫ ∫ = f (x) + g(x) ... (2) Thus, in view of Property (II), it follows by (1) and (2) that ( ) ( ) ( ) f x g x dx + ∫ = ( ) ( ) f x dx g x dx + ∫ ∫ . (IV) For any real number k, ( ) ( ) k f x dx k f x dx = ∫ ∫ Proof By the Property (I), ( ) ( ) d k f x dx k f x dx = ∫ . Also ( ) d k f x dx dx ∫ = ( ) = ( ) d k f x dx k f x dx ∫ Therefore, using the Property (II), we have ( ) ( ) k f x dx k f x dx = ∫ ∫ . (V) Properties (III) and (IV) can be generalised to a finite number of functions f1 , f2 , ..., fn and the real numbers, k1 , k2 , ..., kn giving [ ] 1 1 2 2 ( ) ( ) ( ) n n k f x k f x ... k f x dx + + + ∫ = 1 1 2 2 ( ) ( ) ( ) n n k f x dx k f x dx ... k f x dx + + + ∫ ∫ ∫ . To find an anti derivative of a given function, we search intuitively for a function whose derivative is the given function. The search for the requisite function for finding an anti derivative is known as integration by the method of inspection. We illustrate it through some examples.
- 9. INTEGRALS 295 Example 1Write an anti derivative for each of the following functions using the method of inspection: (i) cos 2x (ii) 3x2 + 4x3 (iii) 1 x , x ≠ 0 Solution (i) We look for a function whose derivative is cos 2x. Recall that d dx sin 2x = 2 cos 2x or cos 2x = 1 2 d dx (sin 2x) = 1 sin 2 2 d x dx Therefore, an anti derivative of cos 2x is 1 sin 2 2 x . (ii) We look for a function whose derivative is 3x2 + 4x3. Note that ( ) 3 4 d x x dx + = 3x2 + 4x3 . Therefore, an anti derivative of 3x2 + 4x3 is x3 + x4 . (iii) We know that 1 1 1 (log ) 0 and [log ( )] ( 1) 0 d d x ,x – x – ,x dx x dx – x x = > = = < Combining above, we get ( ) 1 log 0 d x , x dx x = ≠ Therefore, 1 log dx x x = ∫ is one of the anti derivatives of 1 x . Example 2 Find the following integrals: (i) 3 2 1 x – dx x ∫ (ii) 2 3 ( 1) x dx + ∫ (iii) ∫ 3 2 1 ( 2 – ) + ∫ x x e dx x Solution (i) We have 3 2 2 1 – x – dx x dx – x dx x = ∫ ∫ ∫ (by Property V)
- 10. 296 MATHEMATICS = 1 12 1 1 2 C C 1 1 2 1 – x x – – + + + + + + ; C1 , C2 are constants of integration = 2 1 1 2 C C 2 1 – x x – – – + = 2 1 2 1 + C C 2 x – x + = 2 1 + C 2 x x + , where C = C1 – C2 is another constant of integration. ANote From now onwards, we shall write only one constant of integration in the final answer. (ii) We have 2 2 3 3 ( 1) x dx x dx dx + = + ∫ ∫ ∫ = 2 1 3 C 2 1 3 x x + + + + = 5 3 3 C 5 x x + + (iii) We have 3 3 2 2 1 1 ( 2 ) 2 x x x e – dx x dx e dx – dx x x + = + ∫ ∫ ∫ ∫ = 3 1 2 2 – log + C 3 1 2 x x e x + + + = 5 2 2 2 – log + C 5 x x e x + Example 3 Find the following integrals: (i) (sin cos ) x x dx + ∫ (ii) cosec (cosec cot ) x x x dx + ∫ (iii) 2 1 sin cos – x dx x ∫ Solution (i) We have (sin cos ) sin cos x x dx x dx x dx + = + ∫ ∫ ∫ = – cos sin C x x + +
- 11. INTEGRALS 297 (ii) Wehave 2 (cosec (cosec + cot ) cosec cosec cot x x x dx x dx x x dx = + ∫ ∫ ∫ = – cot cosec C x – x + (iii) We have 2 2 2 1 sin 1 sin cos cos cos – x x dx dx – dx x x x = ∫ ∫ ∫ = 2 sec tan sec x dx – x x dx ∫ ∫ = tan sec C x – x + Example 4 Find the anti derivative F of f defined by f (x) = 4x3 – 6, where F (0) = 3 Solution One anti derivative of f (x) is x4 – 6x since 4 ( 6 ) d x – x dx = 4x3 – 6 Therefore, the anti derivative F is given by F(x) = x4 – 6x + C, where C is constant. Given that F(0) = 3, which gives, 3 = 0 – 6 × 0 + C or C = 3 Hence, the required anti derivative is the unique function F defined by F(x) = x4 – 6x + 3. Remarks (i) We see that if F is an anti derivative of f, then so is F + C, where C is any constant. Thus, if we know one anti derivative F of a function f, we can write down an infinite number of anti derivatives of f by adding any constant to F expressed by F(x) + C, C ∈ R. In applications, it is often necessary to satisfy an additional condition which then determines a specific value of C giving unique anti derivative of the given function. (ii) Sometimes, F is not expressible in terms of elementary functions viz., polynomial, logarithmic, exponential, trigonometric functions and their inverses etc. We are therefore blocked for finding ( ) f x dx ∫ . For example, it is not possible to find 2 – x e dx ∫ by inspection since we can not find a function whose derivative is 2 – x e
- 12. 298 MATHEMATICS (iii) Whenthe variable of integration is denoted by a variable other than x, the integral formulae are modified accordingly. For instance 4 1 4 5 1 C C 4 1 5 y y dy y + = + = + + ∫ 7.2.3 Comparison between differentiation and integration 1. Both are operations on functions. 2. Both satisfy the property of linearity, i.e., (i) [ ] 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ( ) d d d k f x k f x k f x k f x dx dx dx + = + (ii) [ ] 1 1 2 2 1 1 2 2 ( ) ( ) ( ) ( ) k f x k f x dx k f x dx k f x dx + = + ∫ ∫ ∫ Here k1 and k2 are constants. 3. Wehavealreadyseenthatallfunctionsarenotdifferentiable.Similarly, all functions are not integrable. We will learn more about nondifferentiable functions and nonintegrable functions in higher classes. 4. The derivative of a function, when it exists, is a unique function. The integral of a function is not so. However, they are unique upto an additive constant, i.e., any two integrals of a function differ by a constant. 5. When a polynomial function P is differentiated, the result is a polynomial whose degree is 1 less than the degree of P.When a polynomial function P is integrated, the result is a polynomial whose degree is 1 more than that of P. 6. We can speak of the derivative at a point. We never speak of the integral at a point, we speak of the integral of a function over an interval on which the integral is defined as will be seen in Section 7.7. 7. The derivative of a function has a geometrical meaning, namely, the slope of the tangent to the corresponding curve at a point. Similarly, the indefinite integral of a function represents geometrically, a family of curves placed parallel to each other having parallel tangents at the points of intersection of the curves of the familywiththelinesorthogonal(perpendicular)totheaxisrepresentingthevariable of integration. 8. The derivative is used for finding some physical quantities like the velocity of a moving particle, when the distance traversed at any time t is known. Similarly, the integral is used in calculating the distance traversed when the velocity at time t is known. 9. Differentiation is a process involving limits.So is integration, as will be seen in Section 7.7.
- 13. INTEGRALS 299 10. Theprocess of differentiation and integration are inverses of each other as discussed in Section 7.2.2 (i). EXERCISE 7.1 Find an anti derivative (or integral) of the following functions by the method of inspection. 1. sin 2x 2. cos 3x 3. e2x 4. (ax + b)2 5. sin 2x – 4 e3x Find the following integrals in Exercises 6 to 20: 6. 3 (4 + 1) x e dx ∫ 7. 2 2 1 (1– ) x dx x ∫ 8. 2 ( ) ax bx c dx + + ∫ 9. 2 (2 ) x x e dx + ∫ 10. 2 1 x – dx x ∫ 11. 3 2 2 5 4 x x – dx x + ∫ 12. 3 3 4 x x dx x + + ∫ 13. 3 2 1 1 x x x – dx x – − + ∫ 14. (1 ) – x x dx ∫ 15. 2 ( 3 2 3) x x x dx + + ∫ 16. (2 3cos ) x x – x e dx + ∫ 17. 2 (2 3sin 5 ) x – x x dx + ∫ 18. sec (sec tan ) x x x dx + ∫ 19. 2 2 sec cosec x dx x ∫ 20. 2 2 – 3sin cos x x ∫ dx. Choose the correct answer in Exercises 21 and 22. 21. The anti derivative of 1 x x + equals (A) 1 1 3 2 1 2 C 3 x x + + (B) 2 2 3 2 1 C 3 2 x x + + (C) 3 1 2 2 2 2 C 3 x x + + (D) 3 1 2 2 3 1 C 2 2 x x + + 22. If 3 4 3 ( ) 4 d f x x dx x = − such that f (2) = 0. Then f (x) is (A) 4 3 1 129 8 x x + − (B) 3 4 1 129 8 x x + + (C) 4 3 1 129 8 x x + + (D) 3 4 1 129 8 x x + −
- 14. 300 MATHEMATICS 7.3 Methodsof Integration In previous section, we discussed integrals of those functions which were readily obtainable from derivatives of some functions. It was based on inspection, i.e., on the search of a function F whose derivative is f which led us to the integral of f. However, this method, which depends on inspection, is not very suitable for many functions. Hence, we need to develop additional techniques or methods for finding the integrals by reducing them into standard forms. Prominent among them are methods based on: 1. Integration by Substitution 2. Integration using Partial Fractions 3. Integration by Parts 7.3.1 Integration by substitution In this section, we consider the method of integration by substitution. The given integral ( ) f x dx ∫ can be transformed into another form by changing the independent variable x to t by substituting x = g (t). Consider I = ( ) f x dx ∫ Put x = g(t) so that dx dt = g′(t). We write dx = g′(t) dt Thus I = ( ) ( ( )) ( ) f x dx f g t g t dt = ′ ∫ ∫ This change of variable formula is one of the important tools available to us in the name of integration by substitution. It is often important to guess what will be the useful substitution. Usually, we make a substitution for a function whose derivative also occurs in the integrand as illustrated in the following examples. Example 5 Integrate the following functions w.r.t. x: (i) sin mx (ii) 2x sin (x2 + 1) (iii) 4 2 tan sec x x x (iv) 1 2 sin (tan ) 1 – x x + Solution (i) We know that derivative of mx is m. Thus, we make the substitution mx = t so that mdx = dt. Therefore, 1 sin sin mx dx t dt m = ∫ ∫ = – 1 m cos t + C = – 1 m cos mx + C
- 15. INTEGRALS 301 (ii) Derivativeof x2 + 1 is 2x. Thus, we use the substitution x2 + 1 = t so that 2x dx = dt. Therefore, 2 2 sin ( 1) sin x x dx t dt + = ∫ ∫ = – cos t + C = – cos (x2 + 1) + C (iii) Derivative of x is 1 2 1 1 2 2 – x x = . Thus, we use the substitution 1 so that giving 2 x t dx dt x = = dx = 2t dt. Thus, 4 2 4 2 tan sec 2 tan sec x x t t t dt dx t x = ∫ ∫ = 4 2 2 tan sec t t dt ∫ Again, we make another substitution tan t = u so that sec2 t dt = du Therefore, 4 2 4 2 tan sec 2 t t dt u du = ∫ ∫ = 5 2 C 5 u + = 5 2 tan C 5 t + (since u = tan t) = 5 2 tan C (since ) 5 x t x + = Hence, 4 2 tan sec x x dx x ∫ = 5 2 tan C 5 x + Alternatively, make the substitution tan x t = (iv) Derivative of 1 2 1 tan 1 – x x = + . Thus, we use the substitution tan–1 x = t so that 2 1 dx x + = dt. Therefore , 1 2 sin (tan ) sin 1 – x dx t dt x = + ∫ ∫ = – cos t + C = – cos (tan–1 x) + C Now, we discuss some important integrals involving trigonometric functions and their standard integrals using substitution technique. These will be used later without reference. (i) ∫tan = log sec + C x dx x We have sin tan cos x x dx dx x = ∫ ∫
- 16. 302 MATHEMATICS Put cosx = t so that sin x dx = – dt Then tan log C log cos C dt x dx – – t – x t = = + = + ∫ ∫ or tan log sec C x dx x = + ∫ (ii) ∫cot = log sin + C x dx x We have cos cot sin x x dx dx x = ∫ ∫ Put sin x = t so that cos x dx = dt Then cot dt x dx t = ∫ ∫ = log C t + = log sin C x + (iii) ∫sec = log sec + tan + C x dx x x We have sec (sec tan ) sec sec + tan x x x x dx dx x x + = ∫ ∫ Put sec x + tan x = t so that sec x (tan x + sec x) dx = dt Therefore, sec log + C = log sec tan C dt x dx t x x t = = + + ∫ ∫ (iv) ∫cosec = log cosec – cot + C x dx x x We have cosec (cosec cot ) cosec (cosec cot ) x x x x dx dx x x + = + ∫ ∫ Put cosec x + cot x = t so that – cosec x (cosec x + cot x) dx = dt So cosec – –log| | – log|cosec cot | C dt x dx t x x t = = = + + ∫ ∫ = 2 2 cosec cot – log C cosec cot x x x x − + − = log cosec cot C x – x + Example 6 Find the following integrals: (i) 3 2 sin cos x x dx ∫ (ii) sin sin ( ) x dx x a + ∫ (iii) 1 1 tan dx x + ∫
- 17. INTEGRALS 303 Solution (i) Wehave 3 2 2 2 sin cos sin cos (sin ) x x dx x x x dx = ∫ ∫ = 2 2 (1– cos ) cos (sin ) x x x dx ∫ Put t = cos x so that dt = – sin x dx Therefore, 2 2 sin cos (sin ) x x x dx ∫ = 2 2 (1– ) t t dt − ∫ = 3 5 2 4 ( – ) C 3 5 t t – t t dt – – = + ∫ = 3 5 1 1 cos cos C 3 5 – x x + + (ii) Put x + a = t. Then dx = dt. Therefore sin sin ( ) sin ( ) sin x t – a dx dt x a t = + ∫ ∫ = sin cos cos sin sin t a – t a dt t ∫ = cos – sin cot a dt a t dt ∫ ∫ = 1 (cos ) (sin ) log sin C a t – a t + = 1 (cos ) ( ) (sin ) log sin ( ) C a x a – a x a + + + = 1 cos cos (sin ) log sin ( ) C sin x a a a – a x a – a + + Hence, sin sin ( ) x dx x a + ∫ = x cos a – sin a log |sin (x + a)| + C, where, C = – C1 sin a + a cos a, is another arbitrary constant. (iii) cos 1 tan cos sin dx x dx x x x = + + ∫ ∫ = 1 (cos + sin + cos – sin ) 2 cos sin x x x x dx x x + ∫
- 18. 304 MATHEMATICS = 1 1cos – sin 2 2 cos sin x x dx dx x x + + ∫ ∫ = 1 C 1 cos sin 2 2 2 cos sin x x – x dx x x + + + ∫ ... (1) Now, consider cos sin I cos sin x – x dx x x = + ∫ Put cos x + sin x = t so that (cos x – sin x) dx = dt Therefore 2 I log C dt t t = = + ∫ = 2 log cos sin C x x + + Putting it in (1), we get 1 2 C C 1 + + log cos sin 1 tan 2 2 2 2 dx x x x x = + + + ∫ = 1 2 C C 1 + log cos sin 2 2 2 2 x x x + + + = 1 2 C C 1 + log cos sin C C 2 2 2 2 x x x , + + = + EXERCISE 7.2 Integrate the functions in Exercises 1 to 37: 1. 2 2 1 x x + 2. ( )2 log x x 3. 1 log x x x + 4. sin sin (cos ) x x 5. sin ( ) cos ( ) ax b ax b + + 6. ax b + 7. 2 x x + 8. 2 1 2 x x + 9. 2 (4 2) 1 x x x + + + 10. 1 x – x 11. 4 x x + , x > 0 12. 1 3 5 3 ( 1) x – x 13. 2 3 3 (2 3 ) x x + 14. 1 (log )m x x , x > 0, 1 ≠ m 15. 2 9 4 x – x 16. 2 3 x e + 17. 2 x x e
- 19. INTEGRALS 305 18. 1 2 1 – tan x e x + 19. 2 2 1 1 x x e– e + 20. 2 2 2 2 x – x x – x e – e e e + 21. tan2 (2x – 3) 22. sec2 (7 – 4x) 23. 1 2 sin 1 – x – x 24. 2cos 3sin 6cos 4sin x – x x x + 25. 2 2 1 cos (1 tan ) x – x 26. cos x x 27. sin 2 cos 2 x x 28. cos 1 sin x x + 29. cot x log sin x 30. sin 1 cos x x + 31. ( )2 sin 1 cos x x + 32. 1 1 cot x + 33. 1 1 tan – x 34. tan sin cos x x x 35. ( )2 1 log x x + 36. ( )2 ( 1) log x x x x + + 37. ( ) 3 1 4 sin tan 1 – x x x8 + Choose the correct answer in Exercises 38 and 39. 38. 10 9 10 10 10 log 10 x e x x dx x + + ∫ equals (A) 10x – x10 + C (B) 10x + x10 + C (C) (10x – x10 )–1 + C (D) log (10x + x10 ) + C 39. 2 2 equals sin cos dx x x ∫ (A) tan x + cot x + C (B) tan x – cot x + C (C) tan x cot x + C (D) tan x – cot 2x + C 7.3.2 Integration using trigonometric identities When the integrand involves some trigonometric functions, we use some known identities to find the integral as illustrated through the following example. Example 7 Find (i) 2 cos x dx ∫ (ii) sin 2 cos 3 x x dx ∫ (iii) 3 sin x dx ∫
- 20. 306 MATHEMATICS Solution (i) Recallthe identity cos 2x = 2 cos2 x – 1, which gives cos2 x = 1 cos 2 2 x + Therefore, 2 cos ∫ x dx = 1 (1+ cos 2 ) 2 x dx ∫ = 1 1 cos 2 2 2 dx x dx + ∫ ∫ = 1 sin 2 C 2 4 x x + + (ii) Recall the identity sin x cos y = 1 2 [sin (x + y) + sin (x – y)] (Why?) Then sin 2 cos 3 ∫ x xdx = 1 sin 5 sin 2 • ∫ ∫ x dx x dx = 1 1 cos 5 cos C 2 5 – x x + + = 1 1 cos 5 cos C 10 2 – x x + + (iii) From the identity sin 3x = 3 sin x – 4 sin3 x, we find that sin3 x = 3sin sin 3 4 x – x Therefore, 3 sin x dx ∫ = 3 1 sin sin 3 4 4 x dx – x dx ∫ ∫ = 3 1 – cos cos 3 C 4 12 x x + + Alternatively, 3 2 sin sin sin x dx x x dx = ∫ ∫ = 2 (1– cos ) sin x x dx ∫ Put cos x = t so that – sin x dx = dt Therefore, 3 sin x dx ∫ = ( ) 2 1 – t dt − ∫ = 3 2 C 3 t – dt t dt – t + = + + ∫ ∫ = 3 1 cos cos C 3 – x x + + Remark It can be shown using trigonometric identities that both answers are equivalent.
- 21. INTEGRALS 307 EXERCISE 7.3 Findthe integrals of the functions in Exercises 1 to 22: 1. sin2 (2x + 5) 2. sin 3x cos 4x 3. cos 2x cos 4x cos 6x 4. sin3 (2x + 1) 5. sin3 x cos3 x 6. sin x sin 2x sin 3x 7. sin 4x sin 8x 8. 1 cos 1 cos – x x + 9. cos 1 cos x x + 10. sin4 x 11. cos4 2x 12. 2 sin 1 cos x x + 13. cos 2 cos 2 cos cos x – x – α α 14. cos sin 1 sin 2 x – x x + 15. tan3 2x sec 2x 16. tan4x 17. 3 3 2 2 sin cos sin cos x x x x + 18. 2 2 cos 2 2sin cos x x x + 19. 3 1 sin cos x x 20. ( )2 cos 2 cos sin x x x + 21. sin – 1 (cos x) 22. 1 cos ( ) cos ( ) x – a x – b Choose the correct answer in Exercises 23 and 24. 23. 2 2 2 2 sin cos is equal to sin cos x x dx x x − ∫ (A) tan x + cot x + C (B) tan x + cosec x + C (C) – tan x + cot x + C (D) tan x + sec x + C 24. 2 (1 ) equals cos ( ) x x e x dx e x + ∫ (A) – cot (exx ) + C (B) tan (xex ) + C (C) tan (ex) + C (D) cot (ex) + C 7.4 Integrals of Some Particular Functions In this section, we mention below some important formulae of integrals and apply them for integrating many other related standard integrals: (1) ∫ 2 2 1 – = log + C 2 + – dx x a a x a x a
- 22. 308 MATHEMATICS (2) ∫2 2 1 + = log + C 2 – – dx a x a a x a x (3) ∫ – 1 2 2 1 tan C dx x = + a a x + a (4) ∫ 2 2 2 2 = log + – +C – dx x x a x a (5) ∫ – 1 2 2 = sin + C – dx x a a x (6) ∫ 2 2 2 2 = log + + + C + dx x x a x a We now prove the above results: (1) We have 2 2 1 1 ( ) ( ) x – a x a x – a = + = 1 ( ) – ( ) 1 1 1 2 ( ) ( ) 2 x a x – a – a x – a x a a x – a x a + = + + Therefore, 2 2 1 2 dx dx dx – a x – a x a x – a = + ∫ ∫ ∫ = [ ] 1 log ( )| log ( )| C 2 | x – a – | x a a + + = 1 log C 2 x – a a x a + + (2) In view of (1) above, we have 2 2 1 1 ( ) ( ) 2 ( ) ( ) – a x a x a a x a x a x + + − = + − = 1 1 1 2a a x a x + − +
- 23. INTEGRALS 309 Therefore, 22 – dx a x ∫ = 1 2 dx dx a a x a x + − + ∫ ∫ = 1 [ log | | log | |] C 2 a x a x a − − + + + = 1 log C 2 a x a a x + + − ANote The technique used in (1) will be explained in Section 7.5. (3) Put x = a tan θ. Then dx = a sec2 θ dθ. Therefore, 2 2 dx x a + ∫ = 2 2 2 2 θ θ θ sec tan a d a a + ∫ = 1 1 1 1 θ θ C tan C – x d a a a a = + = + ∫ (4) Let x = a secθ. Then dx = a secθ tanθ dθ. Therefore, 2 2 dx x a − ∫ = 2 2 2 secθ tanθ θ sec θ a d a a − ∫ = 1 secθ θ log secθ + tanθ + C d = ∫ = 2 1 2 log 1 C x x – a a + + = 2 2 1 log log C x x – a a + − + = 2 2 log + C x x – a + , whereC = C1 – log|a| (5) Let x = a sinθ. Then dx = a cosθ dθ. Therefore, 2 2 dx a x − ∫ = 2 2 2 θ θ θ cos sin a d a – a ∫ = 1 θ = θ + C = sin C – x d a + ∫ (6) Let x = a tanθ. Then dx = a sec2θ dθ. Therefore, 2 2 dx x a + ∫ = 2 2 2 2 θ θ θ sec tan a d a a + ∫ = 1 θ θ secθ θ = log (sec tan ) C d + + ∫
- 24. 310 MATHEMATICS = 2 1 2 log 1C x x a a + + + = 2 1 log log C x x a |a| 2 + + − + = 2 log C x x a2 + + + , where C = C1 – log |a| Applying these standard formulae, we now obtain some more formulae which are useful from applications point of view and can be applied directly to evaluate other integrals. (7) To find the integral 2 dx ax bx c + + ∫ , we write ax2 + bx + c = 2 2 2 2 2 4 b c b c b a x x a x – a a a a a + + = + + Now, put 2 b x t a + = so that dx = dt and writing 2 2 2 4 c b – k a a = ± . We find the integral reduced to the form 2 2 1 dt a t k ± ∫ depending upon the sign of 2 2 4 c b – a a and hence can be evaluated. (8) To find the integral of the type 2 dx ax bx c + + ∫ , proceeding as in (7), we obtain the integral using the standard formulae. (9) To find the integral of the type 2 px q dx ax bx c + + + ∫ , where p, q, a, b, c are constants, we are to find real numbers A, B such that 2 + = A ( ) + B = A (2 ) + B d px q ax bx c ax b dx + + + To determine A and B, we equate from both sides the coefficients of x and the constant terms. A and B are thus obtained and hence the integral is reduced to one of the known forms.
- 25. INTEGRALS 311 (10) Forthe evaluation of the integral of the type 2 ( ) px q dx ax bx c + + + ∫ , we proceed as in (9) and transform the integral into known standard forms. Let us illustrate the above methods by some examples. Example 8 Find the following integrals: (i) 2 16 dx x − ∫ (ii) 2 2 dx x x − ∫ Solution (i) We have 2 2 2 16 4 dx dx x x – = − ∫ ∫ = 4 log C 8 4 x – x 1 + + [by 7.4 (1)] (ii) ( ) 2 2 2 1 1 = − ∫ ∫ dx dx x x – x – Put x – 1 = t. Then dx = dt. Therefore, 2 2 dx x x − ∫ = 2 1 dt – t ∫ = 1 sin ( ) C – t + [by 7.4 (5)] = 1 sin ( –1) C – x + Example 9 Find the following integrals : (i) 2 6 13 dx x x − + ∫ (ii) 2 3 13 10 dx x x + − ∫ (iii) 2 5 2 dx x x − ∫ Solution (i) We have x2 – 6x + 13 = x2 – 6x + 32 – 32 + 13 = (x – 3)2 + 4 So, 6 13 dx x x 2 − + ∫ = ( )2 2 1 3 2 dx x – + ∫ Let x – 3 = t. Then dx = dt Therefore, 6 13 dx x x 2 − + ∫ = 1 2 2 1 tan C 2 2 2 – dt t t = + + ∫ [by 7.4 (3)] = 1 1 3 tan C 2 2 – x – +
- 26. 312 MATHEMATICS (ii) Thegiven integral is of the form 7.4 (7). We write the denominator of the integrand, 2 3 13 10 x x – + = 2 13 10 3 3 3 x x – + = 2 2 13 17 3 6 6 x – + (completing the square) Thus 3 13 10 dx x x 2 + − ∫ = 2 2 1 3 13 17 6 6 dx x + − ∫ Put 13 6 x t + = . Then dx = dt. Therefore, 3 13 10 dx x x 2 + − ∫ = 2 2 1 3 17 6 dt t − ∫ = 1 17 1 6 log C 17 17 3 2 6 6 t – t + × × + [by 7.4 (i)] = 1 13 17 1 6 6 log C 13 17 17 6 6 x – x + + + + = 1 1 6 4 log C 17 6 30 x x − + + = 1 1 3 2 1 1 log C log 17 5 17 3 x x − + + + = 1 3 2 log C 17 5 x x − + + , where C = 1 1 1 C log 17 3 +
- 27. INTEGRALS 313 (iii) Wehave 2 2 5 2 5 5 dx dx x x x x – 2 = − ∫ ∫ = 2 2 1 5 1 1 5 5 dx x – – ∫ (completing the square) Put 1 5 x – t = . Then dx = dt. Therefore, 5 2 dx x x 2 − ∫ = 2 2 1 5 1 5 dt t – ∫ = 2 2 1 1 log C 5 5 t t – + + [by 7.4 (4)] = 2 1 1 2 log C 5 5 5 x x – x – + + Example 10 Find the following integrals: (i) 2 2 6 5 x dx x x 2 + + + ∫ (ii) 2 3 5 4 x dx x x + − + ∫ Solution (i) Using the formula 7.4 (9), we express x + 2 = ( ) 2 A 2 6 5 B d x x dx + + + = A (4 6) B x + + Equating the coefficients of x and the constant terms from both sides, we get 4A = 1 and 6A + B = 2 or A = 1 4 and B = 1 2 . Therefore, 2 2 6 5 x x x 2 + + + ∫ = 1 4 6 1 4 2 2 6 5 2 6 5 x dx dx x x x x 2 2 + + + + + + ∫ ∫ = 1 2 1 1 I I 4 2 + (say) ... (1)
- 28. 314 MATHEMATICS In I1,put 2x2 + 6x + 5 = t, so that (4x + 6) dx = dt Therefore, I1 = 1 log C dt t t = + ∫ = 2 1 log | 2 6 5| C x x + + + ... (2) and I2 = 2 2 1 5 2 2 6 5 3 2 dx dx x x x x = + + + + ∫ ∫ = 2 2 1 2 3 1 2 2 dx x + + ∫ Put 3 2 x t + = , so that dx = dt, we get I2 = 2 2 1 2 1 2 dt t + ∫ = 1 2 1 tan 2 C 1 2 2 – t + × [by 7.4 (3)] = 1 2 3 tan 2 + C 2 – x + = ( ) 1 2 tan 2 3 + C – x + ... (3) Using (2) and (3) in (1), we get ( ) 2 1 2 1 1 log 2 6 5 tan 2 3 C 4 2 2 6 5 – x dx x x x x x 2 + = + + + + + + + ∫ where, C = 1 2 C C 4 2 + (ii) This integral is of the form given in 7.4 (10). Let us express x + 3 = 2 A (5 4 ) + B d – x – x dx = A (– 4 – 2x) + B Equating the coefficients of x and the constant terms from both sides, we get – 2A = 1 and – 4 A + B = 3, i.e., A = 1 2 – and B = 1
- 29. INTEGRALS 315 Therefore, 2 3 54 x dx x x + − − ∫ = ( ) 2 2 4 2 1 2 5 4 5 4 – – x dx dx – x x x x + − − − − ∫ ∫ = 1 2 – I1 + I2 ... (1) In I1 , put 5 – 4x – x2 = t, so that (– 4 – 2x) dx = dt. Therefore, I1 = ( ) 2 4 2 5 4 – x dx dt t x x − = − − ∫ ∫ = 1 2 C t + = 2 1 2 5 4 C – x – x + ... (2) Now consider I2 = 2 2 5 4 9 ( 2) dx dx x x – x = − − + ∫ ∫ Put x + 2 = t, so that dx = dt. Therefore, I2 = 1 2 2 2 sin + C 3 3 – dt t t = − ∫ [by 7.4 (5)] = 1 2 2 sin C 3 – x + + ... (3) Substituting (2) and (3) in (1), we obtain 2 1 2 3 2 5 – 4 – + sin C 3 5 4 – x x – x x – x – x + + = + ∫ , where 1 2 C C C 2 – = EXERCISE 7.4 Integrate the functions in Exercises 1 to 23. 1. 2 6 3 1 x x + 2. 2 1 1 4x + 3. ( )2 1 2 1 – x + 4. 2 1 9 25 – x 5. 4 3 1 2 x x + 6. 2 6 1 x x − 7. 2 1 1 x – x – 8. 2 6 6 x x a + 9. 2 2 sec tan 4 x x +
- 30. 316 MATHEMATICS 10. 2 1 22 x x + + 11. 2 1 9 6 5 x x + + 12. 2 1 7 6 – x – x 13. ( )( ) 1 1 2 x – x – 14. 2 1 8 3x – x + 15. ( )( ) 1 x – a x –b 16. 2 4 1 2 3 x x x – + + 17. 2 2 1 x x – + 18. 2 5 2 1 2 3 x x x − + + 19. ( )( ) 6 7 5 4 x x – x – + 20. 2 2 4 x x – x + 21. 2 2 2 3 x x x + + + 22. 2 3 2 5 x x – x + − 23. 2 5 3 4 10 x x x + + + . Choose the correct answer in Exercises 24 and 25. 24. 2 equals 2 2 dx x x + + ∫ (A) x tan–1 (x + 1) + C (B) tan–1 (x + 1) + C (C) (x + 1) tan–1 x + C (D) tan–1 x + C 25. 2 equals 9 4 dx x x − ∫ (A) –1 1 9 8 sin C 9 8 x − + (B) –1 1 8 9 sin C 2 9 x − + (C) –1 1 9 8 sin C 3 8 x − + (D) –1 1 9 8 sin C 2 9 x − + 7.5 Integration by Partial Fractions Recall that a rational function is defined as the ratio of two polynomials in the form P( ) Q( ) x x , where P (x) and Q(x) are polynomials in x and Q(x) ≠ 0. If the degree of P(x) is less than the degree of Q(x), then the rational function is called proper, otherwise, it is called improper.The improper rational functions can be reduced to the proper rational
- 31. INTEGRALS 317 functions bylong division process.Thus, if P( ) Q( ) x x is improper, then 1 P ( ) P( ) T( ) Q( ) Q( ) x x x x x = + , where T(x) is a polynomial in x and 1 P ( ) Q( ) x x is a proper rational function. As we know how to integrate polynomials, the integration of any rational function is reduced to the integration of a proper rational function. The rational functions which we shall consider here for integration purposes will be those whose denominators can be factorised into linear and quadratic factors. Assume that we want to evaluate P( ) Q( ) x dx x ∫ , where P( ) Q( ) x x is proper rational function. It is always possible to write the integrand as a sum of simpler rational functions by a method called partial fraction decomposition. After this, the integration can be carried out easily using the already known methods. The following Table 7.2 indicates the types of simpler partial fractions that are to be associated with various kind of rational functions. Table 7.2 S.No. Form of the rational function Form of the partial fraction 1. ( – ) ( – ) px q x a x b + , a ≠ b A B x – a x – b + 2. 2 ( – ) px q x a + ( )2 A B x – a x – a + 3. 2 ( – ) ( ) ( ) px qx r x a x – b x – c + + A B C x – a x – b x –c + + 4. 2 2 ( – ) ( ) px qx r x a x – b + + 2 A B C ( ) x – a x – b x – a + + 5. 2 2 ( – ) ( ) px qx r x a x bx c + + + + 2 A B + C x x – a x bx c + + + , where x2 + bx + c cannot be factorised further In the above table, A, B and C are real numbers to be determined suitably.
- 32. 318 MATHEMATICS Example 11Find ( 1) ( 2) dx x x + + ∫ Solution The integrand is a proper rational function. Therefore, by using the form of partial fraction [Table 7.2 (i)], we write 1 ( 1) ( 2) x x + + = A B 1 2 x x + + + ... (1) where, real numbers A and B are to be determined suitably. This gives 1 = A (x + 2) + B (x + 1). Equating the coefficients of x and the constant term, we get A + B = 0 and 2A + B = 1 Solving these equations, we getA =1 and B = – 1. Thus, the integrand is given by 1 ( 1) ( 2) x x + + = 1 –1 1 2 x x + + + Therefore, ( 1) ( 2) dx x x + + ∫ = 1 2 dx dx – x x + + ∫ ∫ = log 1 log 2 C x x + − + + = 1 log C 2 x x + + + Remark The equation (1) above is an identity, i.e. a statement true for all (permissible) values of x. Some authors use the symbol ‘≡’ to indicate that the statement is an identity and use the symbol ‘=’ to indicate that the statement is an equation, i.e., to indicate that the statement is true only for certain values of x. Example 12 Find 2 2 1 5 6 x dx x x + − + ∫ Solution Here the integrand 2 2 1 5 6 x x – x + + is not proper rational function, so we divide x2 + 1 by x2 – 5x + 6 and find that
- 33. INTEGRALS 319 2 2 1 5 6 x x– x + + = 2 5 5 5 5 1 1 ( 2) ( 3) 5 6 x – x – x – x – x – x + = + + Let 5 5 ( 2) ( 3) x – x – x – = A B 2 3 x – x – + So that 5x – 5 = A (x – 3) + B (x – 2) Equating the coefficients of x and constant terms on both sides, we getA + B = 5 and 3A + 2B = 5. Solving these equations, we get A = – 5 and B = 10 Thus, 2 2 1 5 6 x x – x + + = 5 10 1 2 3 x – x – − + Therefore, 2 2 1 5 6 x dx x – x + + ∫ = 1 5 10 2 3 dx dx dx x – x – − + ∫ ∫ ∫ = x – 5 log |x – 2| + 10 log |x – 3| + C. Example 13 Find 2 3 2 ( 1) ( 3) x dx x x − + + ∫ Solution The integrand is of the type as given in Table 7.2 (4). We write 2 3 2 ( 1) ( 3) x – x x + + = 2 A B C 1 3 ( 1) x x x + + + + + So that 3x – 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2 = A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1 ) Comparing coefficient of x2, x and constant term on both sides, we get A + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = – 2. Solving these equations, we get 11 5 11 A B and C 4 2 4 – – , = = = . Thus the integrand is given by 2 3 2 ( 1) ( 3) x x x − + + = 2 11 5 11 4( 1) 4( 3) 2( 1) – – x x x + + + Therefore, 2 3 2 ( 1) ( 3) x x x − + + ∫ = 2 11 5 11 4 1 2 4 3 ( 1) dx dx dx – x x x − + + + ∫ ∫ ∫ = 11 5 11 log +1 log 3 C 4 2( + 1) 4 x x x + − + + = 11 +1 5 log +C 4 +3 2 ( +1) x x x +
- 34. 320 MATHEMATICS Example 14Find 2 2 2 ( 1) ( 4) x dx x x + + ∫ Solution Consider 2 2 2 ( 1) ( 4) x x x + + and put x2 = y. Then 2 2 2 ( 1) ( 4) x x x + + = ( 1) ( 4) y y y + + Write ( 1) ( 4) y y y + + = A B 1 4 y y + + + So that y = A (y + 4) + B (y + 1) Comparing coefficients of y and constant terms on both sides, we get A + B = 1 and 4A + B = 0, which give A = 1 4 and B 3 3 − = Thus, 2 2 2 ( 1) ( 4) x x x + + = 2 2 1 4 3( 1) 3 ( 4) – x x + + + Therefore, 2 2 2 ( 1) ( 4) x dx x x + + ∫ = 2 2 1 4 3 3 1 4 dx dx – x x + + + ∫ ∫ = 1 1 1 4 1 tan tan C 3 3 2 2 – – x – x + × + = 1 1 1 2 tan tan C 3 3 2 – – x – x + + In the above example, the substitution was made only for the partial fraction part and not for the integration part. Now, we consider an example, where the integration involves a combination of the substitution method and the partial fraction method. Example 15 Find ( ) 2 3 sin 2 cos 5 cos 4sin – d – – φ φ φ φ φ ∫ Solution Let y = sinφ Then dy = cosφ dφ
- 35. INTEGRALS 321 Therefore, ( ) 2 3sin2 cos 5 cos 4sin – d – – φ φ φ φ φ ∫ = 2 (3 – 2) 5 (1 ) 4 y dy – – y – y ∫ = 2 3 2 4 4 y – dy y – y + ∫ = ( )2 3 2 I (say) 2 y – y – = ∫ Now, we write ( )2 3 2 2 y – y – = 2 A B 2 ( 2) y y + − − [by Table 7.2 (2)] Therefore, 3y – 2 = A (y – 2) + B Comparing the coefficients of y and constant term, we get A = 3 and B – 2A = – 2, which gives A = 3 and B = 4. Therefore, the required integral is given by I = 2 3 4 [ + ] 2 ( 2) dy y – y – ∫ = 2 3 + 4 2 ( 2) dy dy y – y – ∫ ∫ = 1 3log 2 4 C 2 y – y − + + − = 4 3log sin 2 C 2 sin – φ − + + φ = 4 3log (2 sin ) + C 2 sin − φ + − φ (since, 2 – sinφ is always positive) Example 16 Find 2 2 1 ( 2) ( 1) x x dx x x + + + + ∫ Solution The integrand is a proper rational function. Decompose the rational function into partial fraction [Table 2.2(5)]. Write 2 2 1 ( 1) ( 2) x x x x + + + + = 2 A B + C 2 ( 1) x x x + + + Therefore, x2 + x + 1 = A (x2 + 1) + (Bx + C) (x + 2)
- 36. 322 MATHEMATICS Equating thecoefficients of x2 , x and of constant term of both sides, we get A + B =1, 2B + C = 1 and A + 2C = 1. Solving these equations, we get 3 2 1 A , B and C 5 5 5 = = = Thus, the integrand is given by 2 2 1 ( 1) ( 2) x x x x + + + + = 2 2 1 3 5 5 5 ( 2) 1 x x x + + + + = 2 3 1 2 1 5 ( 2) 5 1 x x x + + + + Therefore, 2 2 1 ( +1) ( 2) x x dx x x + + + ∫ = 2 2 3 1 2 1 1 5 2 5 5 1 1 dx x dx dx x x x + + + + + ∫ ∫ ∫ = 2 1 3 1 1 log 2 log 1 tan C 5 5 5 – x x x + + + + + EXERCISE 7.5 Integrate the rational functions in Exercises 1 to 21. 1. ( 1) ( 2) x x x + + 2. 2 1 9 x – 3. 3 1 ( 1) ( 2) ( 3) x – x – x – x – 4. ( 1) ( 2) ( 3) x x – x – x – 5. 2 2 3 2 x x x + + 6. 2 1 (1 2 ) – x x – x 7. 2 ( 1) ( – 1) x x x + 8. 2 ( 1) ( 2) x x – x + 9. 3 2 3 5 1 x x – x x + − + 10. 2 2 3 ( 1) (2 3) x x – x − + 11. 2 5 ( 1) ( 4) x x x + − 12. 3 2 1 1 x x x + + − 13. 2 2 (1 ) (1 ) x x − + 14. 2 3 1 ( 2) x – x + 15. 4 1 1 x − 16. 1 ( 1) n x x + [Hint: multiply numerator and denominator byx n – 1 and put xn = t ] 17. cos (1– sin ) (2 – sin ) x x x [Hint : Put sin x = t]
- 37. INTEGRALS 323 18. 2 2 22 ( 1) ( 2) ( 3)( 4) x x x x + + + + 19. 2 2 2 ( 1) ( 3) x x x + + 20. 4 1 ( 1) x x – 21. 1 ( 1) x e – [Hint : Put ex = t] Choose the correct answer in each of the Exercises 22 and 23. 22. ( 1) ( 2) x dx x x − − ∫ equals (A) 2 ( 1) log C 2 x x − + − (B) 2 ( 2) log C 1 x x − + − (C) 2 1 log C 2 x x − + − (D) log ( 1) ( 2) C x x − − + 23. 2 ( 1) dx x x + ∫ equals (A) 2 1 log log ( +1) + C 2 x x − (B) 2 1 log log ( +1) + C 2 x x + (C) 2 1 log log ( +1) +C 2 x x − + (D) 2 1 log log ( +1) + C 2 x x + 7.6 Integration by Parts In this section, we describe one more method of integration, that is found quite useful in integrating products of functions. If u and v are any two differentiable functions of a single variable x (say). Then, by the product rule of differentiation, we have ( ) d uv dx = dv du u v dx dx + Integrating both sides, we get uv = dv du u dx v dx dx dx + ∫ ∫ or dv u dx dx ∫ = du uv – v dx dx ∫ ... (1) Let u = f (x) and dv dx = g(x). Then du dx = f ′(x) and v = ( ) g x dx ∫
- 38. 324 MATHEMATICS Therefore, expression(1) can be rewritten as ( ) ( ) f x g x dx ∫ = ( ) ( ) [ ( ) ] ( ) f x g x dx – g x dx f x dx ′ ∫ ∫ ∫ i.e., ( ) ( ) f x g x dx ∫ = ( ) ( ) [ ( ) ( ) ] f x g x dx – f x g x dx dx ′ ∫ ∫ ∫ If we take f as the first function and g as the second function, then this formula may be stated as follows: “The integral of the product of two functions = (first function) × (integral of the second function) – Integral of [(differential coefficient of the first function) × (integral of the second function)]” Example 17 Find cos x x dx ∫ Solution Put f (x) = x (first function) and g (x) = cos x (second function). Then, integration by parts gives cos x x dx ∫ = cos [ ( ) cos ] d x x dx – x x dx dx dx ∫ ∫ ∫ = sin sin x x – x dx ∫ = x sin x + cos x + C Suppose, we take f (x) = cos x and g(x) = x. Then cos x x dx ∫ = cos [ (cos ) ] d x x dx – x x dx dx dx ∫ ∫ ∫ = ( ) 2 2 cos sin 2 2 x x x x dx + ∫ Thus, it shows that the integral cos x x dx ∫ is reduced to the comparatively more complicated integral having more power of x. Therefore, the proper choice of the first function and the second function is significant. Remarks (i) It is worth mentioning that integration by parts is not applicable to product of functions in all cases. For instance, the method does not work for sin x x dx ∫ . The reason is that there does not exist any function whose derivative is x sin x. (ii) Observe that while finding the integral of the second function, we did not add any constant of integration. If we write the integral of the second function cos x
- 39. INTEGRALS 325 as sinx + k, where k is any constant, then cos x x dx ∫ = (sin ) (sin ) x x k x k dx + − + ∫ = (sin ) (sin x x k x dx k dx + − − ∫ ∫ = (sin ) cos C x x k x – kx + − + = sin cos C x x x + + This shows that adding a constant to the integral of the second function is superfluous so far as the final result is concerned while applying the method of integration by parts. (iii) Usually, if any function is a power of x or a polynomial in x, then we take it as the first function. However, in cases where other function is inverse trigonometric function or logarithmic function, then we take them as first function. Example 18 Find log x dx ∫ Solution To start with, we are unable to guess a function whose derivative is log x. We take log x as the first function and the constant function 1 as the second function. Then, the integral of the second function is x. Hence, (log .1) x dx ∫ = log 1 [ (log ) 1 ] d x dx x dx dx dx − ∫ ∫ ∫ = 1 (log ) – log C x x x dx x x – x x ⋅ = + ∫ . Example 19 Find x x e dx ∫ Solution Take first function as x and second function as ex. The integral of the second function is ex . Therefore, x x e dx ∫ = 1 x x x e e dx − ⋅ ∫ = xex – ex + C. Example 20 Find 1 2 sin 1 – x x dx x − ∫ Solution Let first function be sin – 1 x and second function be 2 1 x x − . First we find the integral of the second function, i.e., 2 1 x dx x − ∫ . Put t =1 – x2 . Then dt = – 2x dx
- 40. 326 MATHEMATICS Therefore, 2 1 x dx x − ∫= 1 2 dt – t ∫ = 2 – 1 t x = − − Hence, 1 2 sin 1 – x x dx x − ∫ = ( ) 1 2 2 2 1 (sin ) 1 ( 1 ) 1 – x – x – x dx x − − − − ∫ = 2 1 1 sin C – x x x − − + + = 2 1 1 sin C x – x x − − + Alternatively, this integral can also be worked out by making substitution sin–1x = θand then integrating by parts. Example 21 Find sin x e x dx ∫ Solution Take ex as the first function and sin x as second function. Then, integrating by parts, we have I sin ( cos ) cos x x x e x dx e – x e x dx = = + ∫ ∫ = – ex cos x + I1 (say) ... (1) Taking ex and cos x as the first and second functions, respectively, in I1, we get I1 = sin sin x x e x – e x dx ∫ Substituting the value of I1 in (1), we get I = – ex cos x + ex sin x – I or 2I = ex (sin x – cos x) Hence, I = sin (sin cos ) + C 2 x x e e x dx x – x = ∫ Alternatively, above integral can also be determined by taking sin x as the first function and ex the second function. 7.6.1 Integral of the type [ ( ) + ( )] x e f x f x dx ′ ∫ We have I = [ ( ) + ( )] x e f x f x dx ′ ∫ = ( ) + ( ) x x e f x dx e f x dx ′ ∫ ∫ = 1 1 I ( ) , where I = ( ) x x e f x dx e f x dx ′ + ∫ ∫ ... (1) Taking f(x) and ex as the first function and second function, respectively, in I1 and integrating it by parts, we have I1 = f (x) ex – ( ) C x f x e dx ′ + ∫ Substituting I1 in (1), we get I = ( ) ( ) ( ) C x x x e f x f x e dx e f x dx ′ ′ − + + ∫ ∫ = ex f (x) + C
- 41. INTEGRALS 327 Thus, ′ ∫[ ( ) ( )] x e f x + f x dx = ( ) C x e f x + Example 22 Find (i) 1 2 1 (tan ) 1 x – e x x + + ∫ dx (ii) 2 2 ( + 1) ( + 1) x x e x ∫ dx Solution (i) We have I = 1 2 1 (tan ) 1 x – e x dx x + + ∫ Consider f (x) = tan– 1 x, then f ′(x) = 2 1 1 x + Thus, the given integrand is of the form ex [ f (x) + f ′(x)]. Therefore, 1 2 1 I (tan ) 1 x – e x dx x = + + ∫ = ex tan– 1 x + C (ii) We have 2 2 ( +1) I ( +1) x x e x = ∫ dx 2 2 1 +1+1) [ ] ( +1) x x – e dx x = ∫ 2 2 2 1 2 [ ] ( +1) ( +1) x x – e dx x x = + ∫ 2 1 2 [ + ] +1 ( +1) x x – e dx x x = ∫ Consider 1 ( ) 1 x f x x − = + , then 2 2 ( ) ( 1) f x x ′ = + Thus, the given integrand is of the form ex [f (x) + f ′(x)]. Therefore, 2 2 1 1 C 1 ( 1) x x x x e dx e x x + − = + + + ∫ EXERCISE 7.6 Integrate the functions in Exercises 1 to 22. 1. x sin x 2. x sin 3x 3. x2 ex 4. x log x 5. x log 2x 6. x2 log x 7. x sin– 1 x 8. x tan–1 x 9. x cos–1 x 10. (sin–1 x)2 11. 1 2 cos 1 x x x − − 12. x sec2 x 13. tan–1x 14. x (log x)2 15. (x2 + 1) log x
- 42. 328 MATHEMATICS 16. ex (sinx+ cosx) 17. 2 (1 ) x x e x + 18. 1 sin 1 cos x x e x + + 19. 2 1 1 – x e x x 20. 3 ( 3) ( 1) x x e x − − 21. e2x sin x 22. 1 2 2 sin 1 – x x + Choose the correct answer in Exercises 23 and 24. 23. 3 2 x x e dx ∫ equals (A) 3 1 C 3 x e + (B) 2 1 C 3 x e + (C) 3 1 C 2 x e + (D) 2 1 C 2 x e + 24. sec (1 tan ) x e x x dx + ∫ equals (A) ex cos x + C (B) ex sec x + C (C) ex sin x + C (D) ex tan x + C 7.6.2 Integrals of some more types Here, we discuss some special types of standard integrals based on the technique of integration by parts : (i) 2 2 x a dx − ∫ (ii) 2 2 x a dx + ∫ (iii) 2 2 a x dx − ∫ (i) Let 2 2 I x a dx = − ∫ Taking constant function 1 as the second function and integrating by parts, we have I = 2 2 2 2 1 2 2 x x x a x dx x a − − − ∫ = 2 2 2 2 2 x x x a dx x a − − − ∫ = 2 2 2 2 2 2 2 x a a x x a dx x a − + − − − ∫
- 43. INTEGRALS 329 = 2 22 2 2 2 2 dx x x a x a dx a x a − − − − − ∫ ∫ = 2 2 2 2 2 I dx x x a a x a − − − − ∫ or 2I = 2 2 2 2 2 dx x x a a x a − − − ∫ or I = ∫ 2 2 x – a dx = 2 2 2 2 2 – – log + – + C 2 2 x a x a x x a Similarly, integrating other two integrals by parts, taking constant function 1 as the second function, we get (ii) ∫ 2 2 2 2 2 2 2 1 + = + + log + + + C 2 2 a x a dx x x a x x a (iii) ∫ 2 2 2 2 2 –1 1 – = – + sin + C 2 2 a x a x dx x a x a Alternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric substitution x = a secθ in (i), x = a tanθ in (ii) and x = a sinθ in (iii) respectively. Example 23 Find 2 2 5 x x dx + + ∫ Solution Note that 2 2 5 x x dx + + ∫ = 2 ( 1) 4 x dx + + ∫ Put x + 1 = y, so that dx = dy. Then 2 2 5 x x dx + + ∫ = 2 2 2 y dy + ∫ = 2 2 1 4 4 log 4 C 2 2 y y y y + + + + + [using 7.6.2 (ii)] = 2 2 1 ( 1) 2 5 2 log 1 2 5 C 2 x x x x x x + + + + + + + + + Example 24 Find 2 3 2x x dx − − ∫ Solution Note that 2 2 3 2 4 ( 1) x x dx x dx − − = − + ∫ ∫
- 44. 330 MATHEMATICS Put x+ 1 = y so that dx = dy. Thus 2 3 2x x dx − − ∫ = 2 4 y dy − ∫ = 2 1 1 4 4 sin C 2 2 2 – y y y − + + [using 7.6.2 (iii)] = 2 1 1 1 ( 1) 3 2 2 sin C 2 2 – x x x x + + − − + + EXERCISE 7.7 Integrate the functions in Exercises 1 to 9. 1. 2 4 x − 2. 2 1 4x − 3. 2 4 6 x x + + 4. 2 4 1 x x + + 5. 2 1 4x x − − 6. 2 4 5 x x + − 7. 2 1 3x x + − 8. 2 3 x x + 9. 2 1 9 x + Choose the correct answer in Exercises 10 to 11. 10. 2 1 x dx + ∫ is equal to (A) ( ) 2 2 1 1 log 1 C 2 2 x x x x + + + + + (B) 3 2 2 2 (1 ) C 3 x + + (C) 3 2 2 2 (1 ) C 3 x x + + (D) 2 2 2 2 1 1 log 1 C 2 2 x x x x x + + + + + 11. 2 8 7 x x dx − + ∫ is equal to (A) 2 2 1 ( 4) 8 7 9log 4 8 7 C 2 x x x x x x − − + + − + − + + (B) 2 2 1 ( 4) 8 7 9log 4 8 7 C 2 x x x x x x + − + + + + − + + (C) 2 2 1 ( 4) 8 7 3 2log 4 8 7 C 2 x x x x x x − − + − − + − + + (D) 2 2 1 9 ( 4) 8 7 log 4 8 7 C 2 2 x x x x x x − − + − − + − + +
- 45. INTEGRALS 331 7.7 DefiniteIntegral In the previous sections, we have studied about the indefinite integrals and discussed few methods of finding them including integrals of some special functions. In this section, we shall study what is called definite integral of a function. The definite integral has a unique value.A definite integral is denoted by ( ) b a f x dx ∫ , wherea is called the lower limit of the integral and b is called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the interval [a, b], then its value is the difference between the values of F at the end points, i.e., F(b) – F(a). Here, we shall consider these two cases separately as discussed below: 7.7.1 Definite integral as the limit of a sum Let f be a continuous function defined on close interval [a, b]. Assume that all the values taken by the function are non negative, so the graph of the function is a curve above the x-axis. The definite integral ( ) b a f x dx ∫ is the area bounded by the curve y = f (x), the ordinates x = a, x = b and the x-axis.To evaluate this area, consider the region PRSQP between this curve, x-axis and the ordinates x = a and x = b (Fig 7.2). Divide the interval [a, b] into n equal subintervals denoted by [x0 , x1 ], [x1 , x2 ] ,..., [xr – 1 , xr ], ..., [xn – 1 , xn ], where x0 = a, x1 = a + h, x2 = a + 2h, ... , xr = a + rh and xn = b = a + nh or . b a n h − = We note that as n → ∞, h → 0. Fig 7.2 O Y X X' Y' Q P C M D L S A B R a= x0 x1 x2 xr-1 xr x =b n y f x = ( )
- 46. 332 MATHEMATICS The regionPRSQP under consideration is the sum of n subregions, where each subregion is defined on subintervals [xr – 1 , xr ], r = 1, 2, 3, …, n. From Fig 7.2, we have area of the rectangle (ABLC) < area of the region (ABDCA) < area of the rectangle (ABDM) ... (1) Evidently as xr – xr–1 → 0, i.e., h → 0 all the three areas shown in (1) become nearly equal to each other. Now we form the following sums. sn = h [f(x0 ) + … + f (xn - 1 )] = 1 0 ( ) n r r h f x − = ∑ ... (2) and Sn = 1 2 1 [ ( ) ( ) ( )] ( ) n n r r h f x f x f x h f x = + +…+ = ∑ ... (3) Here, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles raised over subintervals [xr–1 , xr ] for r = 1, 2, 3, …, n, respectively. In view of the inequality (1) for an arbitrary subinterval [xr–1 , xr ], we have sn < area of the region PRSQP < Sn ... (4) As n → ∞ strips become narrower and narrower, it is assumed that the limiting values of (2) and (3) are the same in both cases and the common limiting value is the required area under the curve. Symbolically, we write limSn n→∞ = lim n n s →∞ = area of the region PRSQP = ( ) b a f x dx ∫ ... (5) It follows that this area is also the limiting value of any area which is between that of the rectangles below the curve and that of the rectangles above the curve. For the sake of convenience, we shall take rectangles with height equal to that of the curve at the left hand edge of each subinterval. Thus, we rewrite (5) as ( ) b a f x dx ∫ = 0 lim [ ( ) ( ) ... ( ( – 1) ] h h f a f a h f a n h → + + + + + or ( ) b a f x dx ∫ = 1 ( – ) lim [ ( ) ( ) ... ( ( –1) ] n b a f a f a h f a n h n →∞ + + + + + ... (6) where h = – 0 b a as n n → → ∞ The above expression (6) is known as the definition of definite integral as the limit of sum. Remark The value of the definite integral of a function over any particular interval depends on the function and the interval, but not on the variable of integration that we
- 47. INTEGRALS 333 choose torepresent the independent variable. If the independent variable is denoted by t or u instead ofx, we simply write the integral as ( ) b a f t dt ∫ or ( ) b a f u du ∫ instead of ( ) b a f x dx ∫ . Hence, the variable of integration is called a dummy variable. Example 25 Find 2 2 0 ( 1) x dx + ∫ as the limit of a sum. Solution By definition ( ) b a f x dx ∫ = 1 ( – ) lim [ ( ) ( ) ... ( ( –1) ], n b a f a f a h f a n h n →∞ + + + + + where, h = – b a n In this example, a = 0, b = 2, f (x) = x2 + 1, 2 – 0 2 h n n = = Therefore, 2 2 0 ( 1) x dx + ∫ = 1 2 4 2 ( –1) 2 lim [ (0) ( ) ( ) ... ( )] n n f f f f n n n n →∞ + + + + = 2 2 2 2 2 2 1 2 4 (2 – 2) 2 lim [1 ( 1) ( 1) ... 1 ] n n n n n n →∞ + + + + + + + = 2 2 2 - 1 1 2 lim [(1 1 ... 1) (2 4 ... (2 – 2) ] 2 →∞ + + + + + + + 14 4 244 3 n n terms n n n = 2 2 2 2 1 2 2 lim [ (1 2 ... ( –1) ] n n n n n2 →∞ + + + + = 1 4 ( 1) (2 –1) 2 lim [ ] 6 n n n n n n n2 →∞ − + = 1 2 ( 1) (2 –1) 2 lim [ ] 3 n n n n n n →∞ − + = 2 1 1 2 lim [1 (1 ) (2 – )] 3 n n n →∞ + − = 4 2 [1 ] 3 + = 14 3
- 48. 334 MATHEMATICS Example 26Evaluate 2 0 x e dx ∫ as the limit of a sum. Solution By definition 2 0 x e dx ∫ = 2 4 2 – 2 0 1 (2 – 0) lim ... n n n n n e e e e n →∞ + + + + Using the sum to n terms of a G.P., where a = 1, 2 n r e = , we have 2 0 x e dx ∫ = 2 2 1 –1 2 lim [ ] 1 n n n n e n e →∞ − = 2 2 1 –1 2 lim –1 n n e n e →∞ = 2 2 2 ( –1) –1 lim 2 2 n n e e n →∞ ⋅ = e2 – 1 [using 0 ( 1) lim 1 h h e h → − = ] EXERCISE 7.8 Evaluate the following definite integrals as limit of sums. 1. b a x dx ∫ 2. 5 0 ( 1) x dx + ∫ 3. 3 2 2 x dx ∫ 4. 4 2 1 ( ) x x dx − ∫ 5. 1 1 x e dx − ∫ 6. 4 2 0 ( ) x x e dx + ∫ 7.8 Fundamental Theorem of Calculus 7.8.1 Area function We have defined ( ) b a f x dx ∫ as the area of the region bounded by the curve y = f(x), the ordinates x = a and x = b and x-axis. Let x be a given point in [a, b]. Then ( ) x a f x dx ∫ represents the area of the light shaded region Fig 7.3
- 49. INTEGRALS 335 in Fig7.3 [Here it is assumed that f (x) > 0 for x ∈ [a, b], the assertion made below is equally true for other functions as well]. The area of this shaded region depends upon the value of x. In other words, the area of this shaded region is a function of x. We denote this function of x by A(x). We call the function A(x) as Area function and is given by A (x) = ∫ ( ) x a f x dx ... (1) Based on this definition, the two basic fundamental theorems have been given. However, we only state them as their proofs are beyond the scope of this text book. 7.8.2 First fundamental theorem of integral calculus Theorem 1 Let f be a continuous function on the closed interval [a, b] and letA (x) be the area function. Then A′ ′ ′ ′ ′(x) = f (x), for all x ∈ ∈ ∈ ∈ ∈ [a, b]. 7.8.3 Second fundamental theorem of integral calculus We state below an important theorem which enables us to evaluate definite integrals by making use of anti derivative. Theorem 2 Let f be continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then ∫ ( ) b a f x dx = [F( )] = b a x F (b) – F(a). Remarks (i) In words, the Theorem 2 tells us that ( ) b a f x dx ∫ = (value of the anti derivative F of f at the upper limit b – value of the same anti derivative at the lower limit a). (ii) This theorem is very useful, because it gives us a method of calculating the definite integral more easily, without calculating the limit of a sum. (iii) The crucial operation in evaluating a definite integral is that of finding a function whose derivative is equal to the integrand. This strengthens the relationship between differentiation and integration. (iv) In ( ) b a f x dx ∫ , the functionf needs to be well defined and continuous in [a, b]. For instance, the consideration of definite integral 1 3 2 2 2 ( –1) x x dx − ∫ is erroneous since the function f expressed by f(x) = 1 2 2 ( –1) x x is not defined in a portion – 1 < x < 1 of the closed interval [– 2, 3].
- 50. 336 MATHEMATICS Steps forcalculating ( ) b a f x dx ∫ . (i) Find the indefinite integral ( ) f x dx ∫ . Let this be F(x). There is no need to keep integration constant C because if we consider F(x) + C instead of F(x), we get ( ) [F ( ) C] [F( ) C]– [F( ) C] F( ) – F( ) b b a a f x dx x b a b a = + = + + = ∫ . Thus, the arbitrary constant disappears in evaluating the value of the definite integral. (ii) Evaluate F(b) – F(a) = [F ( )]b a x , which is the value of ( ) b a f x dx ∫ . We now consider some examples Example 27 Evaluate the following integrals: (i) 3 2 2 x dx ∫ (ii) 9 3 4 2 2 (30– ) x dx x ∫ (iii) 2 1 ( 1) ( 2) x dx x x + + ∫ (iv) 3 4 0 sin 2 cos 2 t t dt π ∫ Solution (i) Let 3 2 2 I x dx = ∫ . Since 3 2 F ( ) 3 x x dx x = = ∫ , Therefore, by the second fundamental theorem, we get I = 27 8 19 F (3) – F (2) – 3 3 3 = = (ii) Let 9 3 4 2 2 I (30 – ) x dx x = ∫ . We first find the anti derivative of the integrand. Put 3 2 3 30 – . Then – 2 x t x dx dt = = or 2 – 3 x dx dt = Thus, 3 2 2 2 2 – 3 (30 – ) x dt dx t x = ∫ ∫ = 2 1 3 t = 3 2 2 1 F ( ) 3 (30 – ) x x =
- 51. INTEGRALS 337 Therefore, bythe second fundamental theorem of calculus, we have I = 9 3 2 4 2 1 F(9) – F(4) 3 (30 – ) x = = 2 1 1 3 (30 – 27) 30 –8 − = 2 1 1 19 3 3 22 99 − = (iii) Let 2 1 I ( 1) ( 2) x dx x x = + + ∫ Using partial fraction, we get –1 2 ( 1) ( 2) 1 2 x x x x x = + + + + + So ( 1) ( 2) x dx x x + + ∫ = – log 1 2log 2 F( ) x x x + + + = Therefore, by the second fundamental theorem of calculus, we have I = F(2) – F(1) = [– log 3 + 2 log 4] – [– log 2 + 2 log 3] = – 3 log 3 + log 2 + 2 log 4 = 32 log 27 (iv) Let 3 4 0 I sin 2 cos2 t t dt π = ∫ . Consider 3 sin 2 cos2 t t dt ∫ Put sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = 1 2 du So 3 sin 2 cos2 t t dt ∫ = 3 1 2 u du ∫ = 4 4 1 1 [ ] sin 2 F ( )say 8 8 u t t = = Therefore, by the second fundamental theorem of integral calculus I = 4 4 1 1 F ( ) – F (0) [sin –sin 0] 4 8 2 8 π π = =
- 52. 338 MATHEMATICS EXERCISE 7.9 Evaluatethe definite integrals in Exercises 1 to 20. 1. 1 1 ( 1) x dx − + ∫ 2. 3 2 1 dx x ∫ 3. 2 3 2 1 (4 – 5 6 9) x x x dx + + ∫ 4. 4 0 sin 2x dx π ∫ 5. 2 0 cos 2x dx π ∫ 6. 5 4 x e dx ∫ 7. 4 0 tanx dx π ∫ 8. 4 6 cosec x dx π π ∫ 9. 1 0 2 1– dx x ∫ 10. 1 2 0 1 dx x + ∫ 11. 3 2 2 1 dx x − ∫ 12. 2 2 0 cos x dx π ∫ 13. 3 2 2 1 x dx x + ∫ 14. 1 2 0 2 3 5 1 x dx x + + ∫ 15. 2 1 0 x x e dx ∫ 16. 2 2 2 1 5 4 3 x x x + + ∫ 17. 2 3 4 0 (2sec 2) x x dx π + + ∫ 18. 2 2 0 (sin –cos ) 2 2 x x dx π ∫ 19. 2 2 0 6 3 4 x dx x + + ∫ 20. 1 0 ( sin ) 4 x x x e dx π + ∫ Choose the correct answer in Exercises 21 and 22. 21. 3 2 1 1 dx x + ∫ equals (A) 3 π (B) 2 3 π (C) 6 π (D) 12 π 22. 2 3 2 0 4 9 dx x + ∫ equals (A) 6 π (B) 12 π (C) 24 π (D) 4 π 7.9 Evaluation of Definite Integrals by Substitution In the previous sections, we have discussed several methods for finding the indefinite integral. One of the important methods for finding the indefinite integral is the method of substitution.
- 53. INTEGRALS 339 To evaluate( ) b a f x dx ∫ , by substitution, the steps could be as follows: 1. Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce the given integral to a known form. 2. Integrate the new integrand with respect to the new variable without mentioning the constant of integration. 3. Resubstitute for the new variable and write the answer in terms of the original variable. 4. Find the values of answers obtained in (3) at the given limits of integral and find the difference of the values at the upper and lower limits. ANote In order to quicken this method, we can proceed as follows: After performing steps 1, and 2, there is no need of step 3. Here, the integral will be kept in the new variable itself, and the limits of the integral will accordingly be changed, so that we can perform the last step. Let us illustrate this by examples. Example 28 Evaluate 1 4 5 1 5 1 x x dx − + ∫ . Solution Put t = x5 + 1, then dt = 5x4 dx. Therefore, 4 5 5 1 x x dx + ∫ = t dt ∫ = 3 2 2 3 t = 3 5 2 2 ( 1) 3 x + Hence, 1 4 5 1 5 1 x x dx − + ∫ = 1 3 5 2 – 1 2 ( 1) 3 x + = ( ) 3 3 5 5 2 2 2 (1 1) – (–1) 1 3 + + = 3 3 2 2 2 2 0 3 − = 2 4 2 (2 2) 3 3 = Alternatively, first we transform the integral and then evaluate the transformed integral with new limits.
- 54. 340 MATHEMATICS Let t= x5 + 1. Then dt = 5 x4 dx. Note that, when x = – 1, t = 0 and when x = 1, t = 2 Thus, as x varies from – 1 to 1, t varies from 0 to 2 Therefore 1 4 5 1 5 1 x x dx − + ∫ = 2 0 t dt ∫ = 2 3 3 3 2 2 2 0 2 2 2 – 0 3 3 t = = 2 4 2 (2 2) 3 3 = Example 29 Evaluate – 1 1 2 0 tan 1 x dx x + ∫ Solution Let t = tan – 1 x, then 2 1 1 dt dx x = + . The new limits are, when x = 0,t = 0 and when x = 1, 4 t π = . Thus, as x varies from 0 to 1, t varies from 0 to 4 π . Therefore –1 1 2 0 tan 1 x dx x + ∫ = 2 4 4 0 0 2 t t dt π π ∫ = 2 2 1 – 0 2 16 32 π π = EXERCISE 7.10 Evaluate the integrals in Exercises 1 to 8 using substitution. 1. 1 2 0 1 x dx x + ∫ 2. 5 2 0 sin cos d π φ φ φ ∫ 3. 1 – 1 2 0 2 sin 1 x dx x + ∫ 4. 2 0 2 x x + ∫ (Put x + 2 = t2 ) 5. 2 2 0 sin 1 cos x dx x π + ∫ 6. 2 2 0 4 – dx x x + ∫ 7. 1 2 1 2 5 dx x x − + + ∫ 8. 2 2 2 1 1 1 – 2 x e dx x x ∫ Choose the correct answer in Exercises 9 and 10. 9. The value of the integral 1 3 3 1 1 4 3 ( ) x x dx x − ∫ is (A) 6 (B) 0 (C) 3 (D) 4 10. If f (x) = 0 sin x t t dt ∫ , then f′(x) is (A) cosx + x sin x (B) x sinx (C) x cosx (D) sinx + x cosx
- 55. INTEGRALS 341 7.10 SomeProperties of Definite Integrals We list below some important properties of definite integrals.These will be useful in evaluating the definite integrals more easily. P0 : ( ) ( ) b b a a f x dx f t dt = ∫ ∫ P1 : ( ) – ( ) b a a b f x dx f x dx = ∫ ∫ . In particular, ( ) 0 a a f x dx = ∫ P2 : ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx = + ∫ ∫ ∫ P3 : ( ) ( ) b b a a f x dx f a b x dx = + − ∫ ∫ P4 : 0 0 ( ) ( ) a a f x dx f a x dx = − ∫ ∫ (Note that P4 is a particular case of P3 ) P5 : 2 0 0 0 ( ) ( ) (2 ) a a a f x dx f x dx f a x dx = + − ∫ ∫ ∫ P6 : 2 0 0 ( ) 2 ( ) , if (2 ) ( ) a a f x dx f x dx f a x f x = − = ∫ ∫ and 0 if f (2a – x) = – f (x) P7 : (i) 0 ( ) 2 ( ) a a a f x dx f x dx − = ∫ ∫ , if f is an even function, i.e., if f (–x) = f (x). (ii) ( ) 0 a a f x dx − = ∫ , if f is an odd function, i.e., if f (– x) = – f (x). We give the proofs of these properties one by one. Proof of P0 It follows directly by making the substitution x = t. Proof of P1 Let F be anti derivative of f. Then, by the second fundamental theorem of calculus, we have ( ) F ( ) – F ( ) –[F ( ) F ( )] ( ) b a a b f x dx b a a b f x dx = = − = − ∫ ∫ Here, we observe that, if a = b, then ( ) 0 a a f x dx = ∫ . Proof of P2 Let F be anti derivative of f. Then ( ) b a f x dx ∫ = F(b) – F(a) ... (1) ( ) c a f x dx ∫ = F(c) – F(a) ... (2) and ( ) b c f x dx ∫ = F(b) – F(c) ... (3)
- 56. 342 MATHEMATICS Adding (2)and (3), we get ( ) ( ) F( ) – F( ) ( ) c b b a c a f x dx f x dx b a f x dx + = = ∫ ∫ ∫ This proves the property P2 . Proof of P3 Let t = a + b – x. Then dt = – dx. When x = a, t = b and when x = b, t = a. Therefore ( ) b a f x dx ∫ = ( – ) a b f a b t dt − + ∫ = ( – ) b a f a b t dt + ∫ (by P1) = ( – ) b a f a b x + ∫ dx by P0 Proof of P4 Put t = a – x. Then dt = – dx. When x = 0,t = a and when x = a, t = 0. Now proceed as in P3 . Proof of P5 Using P2 , we have 2 2 0 0 ( ) ( ) ( ) a a a a f x dx f x dx f x dx = + ∫ ∫ ∫ . Let t = 2a – x in the second integral on the right hand side. Then dt = – dx. When x = a, t = a and when x = 2a, t = 0.Also x = 2a – t. Therefore, the second integral becomes 2 ( ) a a f x dx ∫ = 0 – (2 – ) a f a t dt ∫ = 0 (2 – ) a f a t dt ∫ = 0 (2 – ) a f a x dx ∫ Hence 2 0 ( ) a f x dx ∫ = 0 0 ( ) (2 ) a a f x dx f a x dx + − ∫ ∫ Proof of P6 Using P5 , we have 2 0 0 0 ( ) ( ) (2 ) a a a f x dx f x dx f a x dx = + − ∫ ∫ ∫ ... (1) Now, if f (2a – x) = f (x), then (1) becomes 2 0 ( ) a f x dx ∫ = 0 0 0 ( ) ( ) 2 ( ) , a a a f x dx f x dx f x dx + = ∫ ∫ ∫ and if f(2a – x) = – f (x), then (1) becomes 2 0 ( ) a f x dx ∫ = 0 0 ( ) ( ) 0 a a f x dx f x dx − = ∫ ∫ Proof of P7 Using P2, we have ( ) a a f x dx − ∫ = 0 0 ( ) ( ) a a f x dx f x dx − + ∫ ∫ . Then Let t = – x in the first integral on the right hand side. dt = – dx. When x = – a, t = a and when x = 0, t = 0. Also x = – t.
- 57. INTEGRALS 343 Therefore () a a f x dx − ∫ = 0 0 – (– ) ( ) a a f t dt f x dx + ∫ ∫ = 0 0 (– ) ( ) a a f x dx f x dx + ∫ ∫ (by P0 ) ... (1) (i) Now, if f is an even function, then f (–x) = f (x) and so (1) becomes 0 0 0 ( ) ( ) ( ) 2 ( ) a a a a a f x dx f x dx f x dx f x dx − = + = ∫ ∫ ∫ ∫ (ii) If f is an odd function, then f (–x) = – f(x) and so (1) becomes 0 0 ( ) ( ) ( ) 0 a a a a f x dx f x dx f x dx − = − + = ∫ ∫ ∫ Example 30 Evaluate 2 3 1 – x x dx − ∫ Solution We note that x3 – x ≥ 0 on [– 1, 0] and x3 – x ≤ 0 on [0, 1] and that x3 – x ≥ 0 on [1, 2]. So by P2 we write 2 3 1 – x x dx − ∫ = 0 1 2 3 3 3 1 0 1 ( – ) – ( – ) ( – ) x x dx x x dx x x dx − + + ∫ ∫ ∫ = 0 1 2 3 3 3 1 0 1 ( – ) ( – ) ( – ) x x dx x x dx x x dx − + + ∫ ∫ ∫ = 0 1 2 4 2 2 4 4 2 –1 0 1 – – – 4 2 2 4 4 2 x x x x x x + + = ( ) 1 1 1 1 1 1 – – – 4 – 2 – – 4 2 2 4 4 2 + + = 1 1 1 1 1 1 – 2 4 2 2 4 4 2 + + − + − + = 3 3 11 2 2 4 4 − + = Example 31 Evaluate 2 4 – 4 sin x dx π π ∫ Solution We observe that sin2 x is an even function. Therefore, by P7 (i), we get 2 4 – 4 sin x dx π π ∫ = 2 4 0 2 sin x dx π ∫
- 58. 344 MATHEMATICS = 4 0 (1cos 2 ) 2 2 x dx π − ∫ = 4 0 (1 cos 2 ) x dx π − ∫ = 4 0 1 – sin 2 2 x x π = 1 1 – sin –0 – 4 2 2 4 2 π π π = Example 32 Evaluate 2 0 sin 1 cos x x dx x π + ∫ Solution Let I = 2 0 sin 1 cos x x dx x π + ∫ . Then, by P4 , we have I = 2 0 ( )sin ( ) 1 cos ( ) x x dx x π π − π − + π − ∫ = 2 0 ( )sin 1 cos x x dx x π π − + ∫ = 2 0 sin I 1 cos x dx x π π − + ∫ or 2 I = 2 0 sin 1 cos x dx x π π + ∫ or I = 2 0 sin 2 1 cos x dx x π π + ∫ Put cos x = t so that – sin x dx = dt. When x = 0, t = 1 and when x = π, t = – 1. Therefore, (by P1 ) we get I = 1 2 1 – 2 1 dt t − π + ∫ = 1 2 1 2 1 dt t − π + ∫ = 1 2 0 1 dt t π + ∫ (by P7 , 2 1 since 1 t + is even function) = 2 1 – 1 – 1 1 0 tan tan 1– tan 0 – 0 4 4 t − π π π = π = π = Example 33 Evaluate 1 5 4 1 sin cos x x dx − ∫ Solution Let I = 1 5 4 1 sin cos x x dx − ∫ . Let f(x) = sin5 x cos4 x. Then f (– x) = sin5 (– x) cos4 (– x) = – sin5 x cos4 x = – f (x), i.e., f is an odd function. Therefore, by P7 (ii), I = 0
- 59. INTEGRALS 345 Example 34Evaluate 4 2 4 4 0 sin sin cos x dx x x π + ∫ Solution Let I = 4 2 4 4 0 sin sin cos x dx x x π + ∫ ... (1) Then, by P4 I = 4 2 0 4 4 sin ( ) 2 sin ( ) cos ( ) 2 2 x dx x x π π − π π − + − ∫ = 4 2 4 4 0 cos cos sin x dx x x π + ∫ ... (2) Adding (1) and (2), we get 2I = 4 4 2 2 2 4 4 0 0 0 sin cos [ ] 2 sin cos x x dx dx x x x π π π + π = = = + ∫ ∫ Hence I = 4 π Example 35 Evaluate 3 6 1 tan dx x π π + ∫ Solution Let I = 3 3 6 6 cos 1 tan cos sin x dx dx x x x π π π π = + + ∫ ∫ ... (1) Then, by P3 I = 3 6 cos 3 6 cos sin 3 6 3 6 x dx x x π π π π + − π π π π + − + + − ∫ = 3 6 sin sin cos x dx x x π π + ∫ ... (2) Adding (1) and (2), we get 2I = [ ] 3 3 6 6 3 6 6 dx x π π π π π π π = = − = ∫ . Hence I 12 π =
- 60. 346 MATHEMATICS Example 36Evaluate 2 0 log sin x dx π ∫ Solution Let I = 2 0 logsinx dx π ∫ Then, by P4 I = 2 2 0 0 log sin log cos 2 x dx x dx π π π − = ∫ ∫ Adding the two values of I, we get 2I = ( ) 2 0 log sin logcos x x dx π + ∫ = ( ) 2 0 log sin cos log2 log2 x x dx π + − ∫ (byaddingandsubtractinglog2) = 2 2 0 0 log sin2 log2 x dx dx π π − ∫ ∫ (Why?) Put 2x = t in the first integral. Then 2 dx = dt, when x = 0, t = 0 and when 2 x π = , t = π. Therefore 2I = 0 1 log sin log2 2 2 t dt π π − ∫ = 2 0 2 log sin log2 2 2 t dt π π − ∫ [by P6 as sin (π – t) = sin t) = 2 0 log sin log2 2 x dx π π − ∫ (by changing variablet to x) = I log2 2 π − Hence 2 0 log sin x dx π ∫ = – log2 2 π .
- 61. INTEGRALS 347 EXERCISE 7.11 Byusing the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 1. 2 2 0 cos x dx π ∫ 2. 2 0 sin sin cos x dx x x π + ∫ 3. 3 2 2 3 3 0 2 2 sin sin cos x dx x x π + ∫ 4. 5 2 5 5 0 cos sin cos x dx x x π + ∫ 5. 5 5 | 2| x dx − + ∫ 6. 8 2 5 x dx − ∫ 7. 1 0 (1 )n x x dx − ∫ 8. 4 0 log (1 tan ) x dx π + ∫ 9. 2 0 2 x x dx − ∫ 10. 2 0 (2log sin logsin2 ) x x dx π − ∫ 11. 2 2 – 2 sin x dx π π ∫ 12. 0 1 sin x dx x π + ∫ 13. 7 2 – 2 sin x dx π π ∫ 14. 2 5 0 cos x dx π ∫ 15. 2 0 sin cos 1 sin cos x x dx x x π − + ∫ 16. 0 log (1 cos ) x dx π + ∫ 17. 0 a x dx x a x + − ∫ 18. 4 0 1 x dx − ∫ 19. Show that 0 0 ( ) ( ) 2 ( ) a a f x g x dx f x dx = ∫ ∫ , if f and g are defined as f(x) = f(a –x) and g(x) + g(a – x) = 4 Choose the correct answer in Exercises 20 and 21. 20. The value of 3 5 2 2 ( cos tan 1) x x x x dx π −π + + + ∫ is (A) 0 (B) 2 (C) π (D) 1 21. The value of 2 0 4 3sin log 4 3cos x dx x π + + ∫ is (A) 2 (B) 3 4 (C) 0 (D) –2
- 62. 348 MATHEMATICS Miscellaneous Examples Example37 Find cos 6 1 sin 6 x x dx + ∫ Solution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx Therefore 1 2 1 cos 6 1 sin 6 6 x x dx t dt + = ∫ ∫ = 3 3 2 2 1 2 1 ( ) C = (1 sin 6 ) C 6 3 9 t x × + + + Example 38 Find 1 4 4 5 ( ) x x dx x − ∫ Solution We have 1 1 4 4 4 3 5 4 1 (1 ) ( ) x x x dx dx x x − − = ∫ ∫ Put – 3 3 4 1 3 1 1– , so that x t dx dt x x − = = = Therefore 1 1 4 4 4 5 ( ) 1 3 x x dx t dt x − = ∫ ∫ = 5 5 4 4 3 1 4 4 1 C = 1 C 3 5 15 t x × + − + Example 39 Find 4 2 ( 1) ( 1) x dx x x − + ∫ Solution We have 4 2 ( 1)( 1) x x x − + = 3 2 1 ( 1) 1 x x x x + + − + − = 2 1 ( 1) ( 1) ( 1) x x x + + − + ... (1) Now express 2 1 ( 1)( 1) x x − + = 2 A B C ( 1) ( 1) x x x + + − + ... (2)
- 63. INTEGRALS 349 So 1= A (x2 + 1) + (Bx + C) (x – 1) = (A + B) x2 + (C – B) x + A – C Equating coefficients on both sides, we get A + B = 0, C – B = 0 and A – C = 1, which give 1 1 A , B C – 2 2 = = = . Substituting values of A, B and C in (2), we get 2 1 ( 1) ( 1) x x − + = 2 2 1 1 1 2( 1) 2 ( 1) 2( 1) x x x x − − − + + ... (3) Again, substituting (3) in (1), we have 4 2 ( 1) ( 1) x x x x − + + = 2 2 1 1 1 ( 1) 2( 1) 2 ( 1) 2( 1) x x x x x + + − − − + + Therefore 4 2 2 – 1 2 1 1 1 log 1 – log ( 1) – tan C 2 2 4 2 ( 1) ( 1) x x dx x x x x x x x = + + − + + − + + ∫ Example 40 Find 2 1 log (log ) (log ) x dx x + ∫ Solution Let 2 1 I log (log ) (log ) x dx x = + ∫ = 2 1 log (log ) (log ) x dx dx x + ∫ ∫ In the first integral, let us take 1 as the second function. Then integrating it by parts, we get I = 2 1 log (log ) log (log ) dx x x x dx x x x − + ∫ ∫ = 2 log (log ) log (log ) dx dx x x x x − + ∫ ∫ ... (1) Again, consider log dx x ∫ , take 1 as the second function and integrate it by parts, we have 2 1 1 – – log log (log ) dx x x dx x x x x = ∫ ∫ ... (2)
- 64. 350 MATHEMATICS Putting (2)in (1), we get 2 2 I log (log ) log (log ) (log ) x dx dx x x x x x = − − + ∫ ∫ = log (log ) C log x x x x − + Example 41 Find cot tan x x dx + ∫ Solution We have I = cot tan x x dx + ∫ tan (1 cot ) x x dx = + ∫ Put tan x = t2 , so that sec2 x dx = 2t dt or dx = 4 2 1 t dt t + Then I = 2 4 1 2 1 (1 ) t t dt t t + + ∫ = 2 2 2 4 2 2 2 1 1 1 1 ( 1) 2 = 2 = 2 1 1 1 2 dt dt t t t dt t t t t t + + + + + − + ∫ ∫ ∫ Put 1 t t − = y, so that 2 1 1 t + dt = dy. Then I = ( ) – 1 – 1 2 2 1 2 2 tan C = 2 tan C 2 2 2 t dy y t y − = + + + ∫ = 2 – 1 – 1 1 tan 1 2 tan C = 2 tan C 2 2tan t x t x − − + + Example 42 Find 4 sin 2 cos 2 9– cos (2 ) x x dx x ∫ Solution Let 4 sin 2 cos 2 I 9 – cos 2 x x dx x = ∫
- 65. INTEGRALS 351 Put cos2 (2x)= t so that 4 sin 2x cos 2x dx = – dt Therefore –1 1 2 2 1 1 1 1 I – – sin C sin cos 2 C 4 4 3 4 3 9 – dt t x t − = = + = − + ∫ Example 43 Evaluate 3 2 1 sin ( ) x x dx − π ∫ Solution Here f (x) = | x sin πx | = sin for 1 1 3 sin for 1 2 x x x x x x π − ≤ ≤ − π ≤ ≤ Therefore 3 2 1 | sin | x x dx − π ∫ = 3 1 2 1 1 sin sin x x dx x x dx − π + − π ∫ ∫ = 3 1 2 1 1 sin sin x x dx x x dx − π − π ∫ ∫ Integrating both integrals on righthand side, we get 3 2 1 | sin | x x dx − π ∫ = 3 1 2 2 2 1 1 – cos sin cos sin x x x x x x − π π − π π + − + π π π π = 2 2 2 1 1 3 1 − − − = + π π π π π Example 44 Evaluate 2 2 2 2 0 cos sin x dx a x b x π + ∫ Solution LetI= 2 2 2 2 2 2 2 2 0 0 ( ) cos sin cos ( ) sin ( ) x dx x dx a x b x a x b x π π π − = + π − + π − ∫ ∫ (usingP4 ) = 2 2 2 2 2 2 2 2 0 0 cos sin cos sin dx x dx a x b x a x b x π π π − + + ∫ ∫ = 2 2 2 2 0 I cos sin dx a x b x π π − + ∫ Thus 2I = 2 2 2 2 0 cos sin dx a x b x π π + ∫
- 66. 352 MATHEMATICS or I= 2 2 2 2 2 2 2 2 2 0 0 2 2 2 cos sin cos sin dx dx a x b x a x b x π π π π = ⋅ + + ∫ ∫ (using P6 ) = 2 4 2 2 2 2 2 2 2 2 0 4 cos sin cos sin π π π π + + + ∫ ∫ dx dx a x b x a x b x = 2 2 2 4 2 2 2 2 2 2 0 4 sec cosec tan cot π π π π + + + ∫ ∫ xdx xdx a b x a x b = ( ) 0 1 2 2 2 2 2 2 0 1 tan t cot π − = = + + ∫ ∫ dt du put x and x u a b t a u b = 1 0 –1 –1 0 1 tan – tan π π bt au ab a ab b = –1 –1 tan tan π + b a ab a b = 2 2 π ab Miscellaneous Exercise on Chapter 7 Integrate the functions in Exercises 1 to 24. 1. 3 1 x x − 2. 1 x a x b + + + 3. 2 1 x ax x − [Hint:Putx = a t ] 4. 3 2 4 4 1 ( 1) x x + 5. 1 1 3 2 1 x x + [Hint: 1 1 1 1 3 2 3 6 1 1 1 x x x x = + + , put x = t6 ] 6. 2 5 ( 1) ( 9) x x x + + 7. sin sin ( ) x x a − 8. 5 log 4 log 3 log 2 log x x x x e e e e − − 9. 2 cos 4 sin x x − 10. 8 8 2 2 sin cos 1 2sin cos x x x − − 11. 1 cos ( ) cos ( ) x a x b + + 12. 3 8 1 x x − 13. (1 ) (2 ) x x x e e e + + 14. 2 2 1 ( 1) ( 4) x x + + 15. cos3x elog sinx 16. e3 logx (x4 + 1)– 1 17. f ′ (ax + b) [f (ax + b)]n 18. 3 1 sin sin ( ) x x + α 19. 1 1 1 1 sin cos sin cos x x x x − − − − − + , x ∈ [0, 1]
- 67. INTEGRALS 353 20. 1 1 x x − + 21. 2 sin2 1 cos2 x x e x + + 22. 2 2 1 ( 1) ( 2) x x x x + + + + 23. – 1 1 tan 1 x x − + 24. 2 2 4 1 log ( 1) 2 log x x x x + + − Evaluate the definite integrals in Exercises 25 to 33. 25. 2 1 sin 1 cos π π − − ∫ x x e dx x 26. 4 4 4 0 sin cos cos sin x x dx x x π + ∫ 27. 2 2 2 2 0 cos cos 4 sin x dx x x π + ∫ 28. 3 6 sin cos sin 2 x x dx x π π + ∫ 29. 1 0 1 dx x x + − ∫ 30. 4 0 sin cos 9 16 sin 2 x x dx x π + + ∫ 31. 1 2 0 sin 2 tan (sin ) x x dx π − ∫ 32. 0 tan sec tan x x dx x x π + ∫ 33. 4 1 [ 1| | 2| | 3|] x x x dx − + − + − ∫ Prove the following (Exercises 34 to 39) 34. 3 2 1 2 2 log 3 3 ( 1) dx x x = + + ∫ 35. 1 0 1 x x e dx = ∫ 36. 1 17 4 1 cos 0 x x dx − = ∫ 37. 3 2 0 2 sin 3 x dx π = ∫ 38. 3 4 0 2 tan 1 log2 x dx π = − ∫ 39. 1 1 0 sin 1 2 x dx − π = − ∫ 40. Evaluate 1 2 3 0 x e dx − ∫ as a limit of a sum. Choose the correct answers in Exercises 41 to 44. 41. x x dx e e− + ∫ is equal to (A) tan–1 (ex) + C (B) tan–1 (e–x) + C (C) log (ex – e–x ) + C (D) log (ex + e–x ) + C 42. 2 cos2 (sin cos ) x dx x x + ∫ is equal to (A) –1 C sin cos x x + + (B) log |sin cos | C x x + + (C) log |sin cos | C x x − + (D) 2 1 (sin cos ) x x +
- 68. 354 MATHEMATICS 43. Iff (a + b – x) = f (x), then ( ) b a x f x dx ∫ is equal to (A) ( ) 2 b a a b f b x dx + − ∫ (B) ( ) 2 b a a b f b x dx + + ∫ (C) ( ) 2 b a b a f x dx − ∫ (D) ( ) 2 b a a b f x dx + ∫ 44. The value of 1 1 2 0 2 1 tan 1 x dx x x − − + − ∫ is (A) 1 (B) 0 (C) –1 (D) 4 π Summary ® Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation. Let F( ) ( ) d x f x dx = . Then we write ( ) F ( ) C f x dx x = + ∫ . These integrals are called indefinite integrals or general integrals, C is called constant of integration.All these integrals differ by a constant. ® From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis. ® Some properties of indefinite integrals are as follows: 1. [ ( ) ( )] ( ) ( ) f x g x dx f x dx g x dx + = + ∫ ∫ ∫ 2. For any real number k, ( ) ( ) k f x dx k f x dx = ∫ ∫ More generally, if f1, f2, f3, ... , fn are functions and k1, k2, ... ,kn are real numbers. Then 1 1 2 2 [ ( ) ( ) ... ( )] n n k f x k f x k f x dx + + + ∫ = 1 1 2 2 ( ) ( ) ... ( ) n n k f x dx k f x dx k f x dx + + + ∫ ∫ ∫
- 69. INTEGRALS 355 ® Somestandard integrals (i) 1 C 1 n n x x dx n + = + + ∫ , n ≠ – 1. Particularly, C dx x = + ∫ (ii) cos sin C x dx x = + ∫ (iii) sin – cos C x dx x = + ∫ (iv) 2 sec tan C x dx x = + ∫ (v) 2 cosec – cot C x dx x = + ∫ (vi) sec tan sec C x x dx x = + ∫ (vii) cosec cot – cosec C x x dx x = + ∫ (viii) 1 2 sin C 1 dx x x − = + − ∫ (ix) 1 2 cos C 1 dx x x − = − + − ∫ (x) 1 2 tan C 1 dx x x − = + + ∫ (xi) 1 2 cot C 1 dx x x − = − + + ∫ (xii) C x x e dx e = + ∫ (xiii) C log x x a a dx a = + ∫ (xiv) 1 2 sec C 1 dx x x x − = + − ∫ (xv) 1 2 cosec C 1 dx x x x − = − + − ∫ (xvi) 1 log| | C dx x x = + ∫ ® Integration by partial fractions Recall that a rational function is ratio of two polynomials of the form P( ) Q( ) x x , where P(x) and Q (x) are polynomials in x and Q (x) ≠ 0. If degree of the polynomial P (x) is greater than the degree of the polynomial Q (x), then we may divide P (x) by Q (x) so that 1 P ( ) P( ) T ( ) Q( ) Q( ) x x x x x = + , where T(x) is a polynomial in x and degree of P1 (x) is less than the degree of Q(x). T(x) being polynomial can be easily integrated. 1 P ( ) Q( ) x x can be integrated by
- 70. 356 MATHEMATICS expressing 1 P( ) Q( ) x x as the sum of partial fractions of the following type: 1. ( ) ( ) px q x a x b + − − = A B x a x b + − − , a ≠ b 2. 2 ( ) px q x a + − = 2 A B ( ) x a x a + − − 3. 2 ( ) ( ) ( ) px qx r x a x b x c + + − − − = A B C x a x b x c + + − − − 4. 2 2 ( ) ( ) px qx r x a x b + + − − = 2 A B C ( ) x a x b x a + + − − − 5. 2 2 ( ) ( ) px qx r x a x bx c + + − + + = 2 A B + C x x a x bx c + − + + where x2 + bx + c can not be factorised further. ® Integration by substitution A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution. When the integrand involves some trigonometric functions, we use some well known identities to find the integrals. Using substitution technique, we obtain the following standard integrals. (i) tan log sec C x dx x = + ∫ (ii) cot log sin C x dx x = + ∫ (iii) sec log sec tan C x dx x x = + + ∫ (iv) cosec log cosec cot C x dx x x = − + ∫ ® Integrals of some special functions (i) 2 2 1 log C 2 dx x a a x a x a − = + + − ∫ (ii) 2 2 1 log C 2 dx a x a a x a x + = + − − ∫ (iii) 1 2 2 1 tan C dx x a a x a − = + + ∫
- 71. INTEGRALS 357 (iv) 22 2 2 log C dx x x a x a = + − + − ∫ (v) 1 2 2 sin C dx x a a x − = + − ∫ (vi) 2 2 2 2 log | | C dx x x a x a = + + + + ∫ ® Integration by parts For given functions f1 and f2 , we have 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) d f x f x dx f x f x dx f x f x dx dx dx ⋅ = − ⋅ ∫ ∫ ∫ ∫ , i.e., the integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}. Care must be taken in choosing the first function and the second function. Obviously, we must take that function as the second function whose integral is well known to us. ® [ ( ) ( )] ( ) C x x e f x f x dx e f x dx ′ + = + ∫ ∫ ® Some special types of integrals (i) 2 2 2 2 2 2 2 log C 2 2 x a x a dx x a x x a − = − − + − + ∫ (ii) 2 2 2 2 2 2 2 log C 2 2 x a x a dx x a x x a + = + + + + + ∫ (iii) 2 2 2 2 2 1 sin C 2 2 x a x a x dx a x a − − = − + + ∫ (iv) Integrals of the types 2 2 or dx dx ax bx c ax bx c + + + + ∫ ∫ can be transformed into standard form by expressing ax2 + bx + c = 2 2 2 2 2 4 b c b c b a x x a x a a a a a + + = + + − (v) Integrals of the types 2 2 or px q dx px q dx ax bx c ax bx c + + + + + + ∫ ∫ can be
- 72. 358 MATHEMATICS transformed intostandard form by expressing 2 A ( ) B A (2 ) B d px q ax bx c ax b dx + = + + + = + + , where Aand B are determined by comparing coefficients on both sides. ® We have defined ( ) b a f x dx ∫ as the area of the region bounded by the curve y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b]. Then ( ) x a f x dx ∫ represents the Area function A (x). This concept of area function leads to the Fundamental Theorems of Integral Calculus. ® First fundamental theorem of integral calculus Let the area function be defined by A(x) = ( ) x a f x dx ∫ for all x ≥ a, where the function f is assumed to be continuous on [a, b]. ThenA′(x) = f (x) for all x ∈ [a, b]. ® Second fundamental theorem of integral calculus Let f be a continuous function of x defined on the closed interval [a, b] and let F be another function such that F( ) ( ) d x f x dx = for all x in the domain of f, then [ ] ( ) F( ) C F ( ) F ( ) b b a a f x dx x b a = + = − ∫ . This is called the definite integral of f over the range [a, b], where a and b are called the limits of integration, a being the lower limit and b the upper limit. —v v v v v—