INTEGRATION-1.pptx

INTEGRATION
PRESENTED BY:
AIML DEPARTMENT
Sayan Sen, University Roll No: 10830621008
Rohan Dwivedi, University Roll No: 10830621013
Sarbasis Mrinal Banerjee, University Roll No: 10830621014
GUIDED BY
Honourable Dr. Animesh Upadhyay Sir
HOD of Mathematics
Asansol Engineering College, Asansol
(IMPROPER INTEGRATION)

contents
01 HISTORY OF INTEGRATION
02 WHAT IS INTEGRATION?
03
HOW DOES INTEGRATION HELPS US
04
05
06
07
WHY DO WE USE INTEGRATION?
TYPES OF INTEGRATION
CONCLUSION
IMPROPER INTEGRATION

01 HISTORY OF INTEGRATION
• Integration is one of the most important mathematical concept ever
conceived throughout the entire history of mathematical research.
• In the framework of calculus, it appears as an indispensable part of any
subcategory in applied mathematics. Is appears seemingly in every field,
along with differentiation, a status as a basic tool of mathematics
understanding.
• Although methods of calculating areas and volumes dated from ancient
Greek mathematics, the principles of integration were formulated
independently by Issac Newton and Gottfried Wilhelm Leibniz in the late
17th century, who thought of the area under a curve as an infinite sum of
rectangles of infinitesimal width.
• Bernhard Riemann later gave a rigorous definition of integration, which is
based on a limiting procedure that approximate the area of a curvline
region by breaking the region into thin vertical slabs.
Sir Isaac Newton
Gottfried Wilhelm Leibniz
Bernhard Riemann

02 What is Integration?
If
-> dx(F(x)) = f(x),
Then
-> ∫f(x)dx = F(x) + c
The function F(x) is called anti-derivative or integral or primitive of the given
function f(x) and c is known as the constant of integration or the arbitrary
constant.
The function f(x) is called the integrand and f(x)dx is known as the element
of integration.
Definition of integration

02 What is Integration?
• Integration is the act of bringing together smaller component into a single
system that functions as one.
• In terms of IT context, we can say that, integration refers to the end result of
a process that aims to stitch together different subsystems so that the data
contained in each become part of a larger, more comprehensive system that,
ideally, quickly and easily shares data when needed.
• This if often required that companies build a customized architecture or
structure of application to combine new or existing hardware, software and
other communications.

03 Why do we use Integration?
• Because it is a powerful problem solving methodology that uncovered a uniform approach to
solving a enormous variety of problems that previously were deemed either extremely
difficult or unattainable although.
• One of its central ideas, that can be called “Division” or “Divide and Conquer”, is an
awesomely beautiful as it is powerful:
• - divide a whole into an arbitrary number of parts each of which is small and simple
enough to be solved on its own.
• - find a way to combine all the solutions of all of the parts into a solution of the whole.

04
Types of Integration
 Integration by Substitution
 Integration by Parts
 Integration using Trigonometric Identities
 Integration of Some particular fraction
 Integration By some Partial Function
 Proper Partial Function
 Improper Partial Function

04 Types of Integration
Integration by Substitution
• We can find the integrating by introducing a new independent
variable when it is difficult to find the integration of a function.
• By changing the independent variable x to t, in a given form of
integral function say ∫f(x).
• Let’s substitute the value of independent x = g(t) in the integral
function ∫f(x),
• We get dx/dt = g’(t)
• Or, dx = g’(t) dt
• Thus, from the above substitution, we get,
• I = ∫f(x) dx = f(g(t).g’(t)) dt

Integration By Part
• If the integrand function can be re[resented as a multiple of two or more
functions, the integrating of any given function can be done by using the
Integration by Parts method.
• Lets us take an integrand function that is equal to f(x)g(x)
• In mathematics here’s how integration by parts is represented.
• ∫f(x)g(x)dx = f(x)∫f(x)dx - ∫(f’(x).∫g(x)dx)dx
• Which can be further written as Integral of the products of any two function
• = (First function x Integral of the second function) - Integral of (differentiation
of the first function) x Integral of the Second Function (differentiation of the
first function) x Integral of the Second Function

Trigonometric identities are used to simplify any integral function which consists of
trigonometric functions. It simplifies the integral function so that it can be easily
integrated.
Integration Using Trigonometric Identities
Integration of some Particular Function
• Many other standard integrals can be integrated using
some important integration formulas.
• Here are the six important formulas listed below -
• -> ∫ dx/ (x2 – a2) = ½ a log | (x – a) / (x + a) | + c
• -> ∫ dx/ (a2 – x2) = ½ a log | (a + x) / (a – x) | + c
• -> ∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + c
• -> ∫ dx /√ (x2 – a2) = log| x+√(x2 – a2) | + c
• -> ∫ dx /√ (a2 – x2) = sin–1 (xa) + c
• -> ∫ dx /√ (x2 + a2) = log | x + √(x2 + a2) | + c
• Where, c = constant

• In mathematics, rational numbers can be expressed in the form of
• p, q & pq where p and q are integers and where the value of the denominator q
is not equal to zero.
• The ratio of two polynomials is known as a rational fraction and it can be
expressed in the form of p(x), q(x) & p(x)q(x), where the value of p(x) should not
be equal to zero.
• The two forms of partial fraction have been described below-
 Proper Partial function: When the degree of the denominator is more than
the degree of the numerator, the function is known as a proper partial
function.
 Improper Partial function: When the degree of the denominator is less
than the degree of the numerator then the fraction is known as improper
partial function. Thus, the fraction can be simplified into parts and can be
integrated easily.
Integrating by Some Partial Function

05
Improper Integration
 What is Improper Integration
 Types of Improper Integration
 Improper Integration of First Kind
 Improper Integration of Second Kind
 qwe

05 Improper Integration
Consider,
This integral is called an improper integral or an infinite integral when either a or b or both are infinite
or f(x) is unbounded in a≤ x ≤ b.
If f(x) is unbounded in [a,b] then there exists at least one point C in [a,b] such that in the modulus of C,
f(x) is not bounded. This point C is called point of infinite discontinuity of the function f(x) or the point
of singularity of the integral (I).
Example: -> Evaluate the following Integral.
Solution:
What is Improper Integration?

05 Improper Integration
Types of Improper Integration
There are two types of Improper Integration.
1. Improper Integral of First Kind.
2. Improper Integral of Second Kind.
 Improper Integral of First Kind
In Improper Integral of First Kind, the limits of integration are
infinite.
There are three types of Improper Integral of First Kind.
 When the upper limit of the improper integral is
infinite.
 When the lower limit of the improper integral is
infinite.
 When both the limit of the improper integral is
infinite.

05 Improper Integration: Improper Integration of first kind
 Improper Integral of First Kind
 Type 1: When the upper limit of the improper integral is infinite.
Let the function f(x) be bounded and integrable in a≤ x ≤ X for all X>a. Then the
improper integral f(x)dx is defined as f(x)dx, provided the limit exists.
Hence f(x)dx = f(x)dx
 Type 2: Let f(x) be bounded and integrable in X≤ x≤ b for every X<b and
f(x)dx exists finitely.
Then the improper integral f(x)dx.
Then f(x)dx = f(x)dx

05 Improper Integration: Improper Integration of first kind
 Type 3:
Consider the improper integral f(x)dx.
Let c be any number. Then we can write
f(x)dx = f(x)dx + f(x)dx
= f(x)dx + f(x)dx, provided limits exists finitely.

05
 Improper Integral of Second Kind
 Type 1:
Let a be the only point of infinite discontinuity of the function f(x). Then the
improper integral f(x)dx is defined as
f(x)dx, provided limit exist.
Hence f(x)ds = f(x)dx
Improper Integration: Improper Integration of second kind

05 Improper Integration: Improper Integration of second kind
 Type 2:
Let b be the only point of infinite discontinuity of the function f(x).
Then the improper integral f(x)dx
is defined as f(x)dx, provided limit exist.
Thus f(x) dx = f(x)dx.
 Type 3:
Let both the end points a and b be the only point og infinite discontinuity of
the function f(x). We take any point c such that a<c<b.
Then the improper integral f(x)dx, can be written as
f(x)dx = f(x)dx + f(x)dx

05 Improper Integration: Improper Integration of second kind
And by type 1 and type 2, we have
f(x)dx = f(x)dx + f(x)dx,
provided both limits exist.
 Type 4:
Let c be the only point of infinite discontinuity of the function f(x) in [a, b] so that
a < c < b. Then we break the integral
f(x) dx into two parts f(x)dx and f(x) dx.
Then by type 1 and type 2 we have
f(x) dx = f(x)dx + f(x)dx,
provided both limits exists.

06 How does Integration Helps Us

05 How does Integration helps us
Some real life examples of Integration
As Integration dated back in 2000 years old. Various architecture marvels had its basis laid by
the fundamentals of Integration.
1. Hoover Dam:
The Hoover Dam is an engineering marvel. When Lake Mead,
the reserviour behind the dam, is full, the dam withstand a
great deal of force.
2. Engineering and Physics:
Definite integrals can be used to determine mass of object if density is unknown.
Work can also be calculated from the integrating a force function, or when countering the force
of Gravity.

05 How does Integration helps us
3. Moments and Center of Mass:
Many of peformers or athelets who spins plates or peform stunts maintain a definite psoture for
smooth running. Mahtmeatically, the sweet spot of the posture is calculated by Integration.
4. Exponential Growth and Decay:
Exponential growth and decay show up in a host of nautral application. From population growth
and continuously compound intrest to radioactive decay and Newton’s law of cooling. All this functions are
eloberated using Integrals.
5. In Medical Science:
Integrals are used to determine the growth of bacterias in the laboratory by keeping variables
such as a change in temperature and foodstuff.
6. In Medicine:
To study the rate of spread of infectious disease, the field of epidemiology uses medical seine to
determine how fast a disease is spreading, its origin, and how to best treat it.
7. In Statistics:
To estimate survey data to help improve marketing plans for different companies because a
survey reqires many different questions with a range of possible answers.

07 Conclusion
The world we see, we live and we admire had all its weight over the basis of Integrals.
From Pharaohs of Egypt who uses integrals for building pyramid to us who are now
heading towards Metaverse in Web 3.0, Integrals was, is and always be a part of it.
Simply we can say that -
“Things grow Stronger when It Integrates.”

INTEGRATION-1.pptx

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INTEGRATION-1.pptx