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- Bioche's rules, formulated by the French mathematician (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines. In the following, is a rational expression in and . In order to calculate , consider the integrand . We consider the behavior of this entire integrand, including the , under translation and reflections of the t axis. The translations and reflections are ones that correspond to the symmetries and periodicities of the basic trigonometric functions. Bioche's rules state that: 1.
* If , a good change of variables is . 2.
* If , a good change of variables is . 3.
* If , a good change of variables is . 4.
* If two of the preceding relations both hold, a good change of variables is . 5.
* In all other cases, use . Because rules 1 and 2 involve flipping the t axis, they flip the sign of dt, and therefore the behavior of ω under these transformations differs from that of ƒ by a sign. Although the rules could be stated in terms of ƒ, stating them in terms of ω has a mnemonic advantage, which is that we choose the change of variables u(t) that has the same symmetry as ω. These rules can be, in fact, stated as a theorem: one shows that the proposed change of variable reduces (if the rule applies and if f is actually of the form ) to the integration of a rational function in a new variable, which can be calculated by partial fraction decomposition. (en)
- En mathématiques, et plus précisément en analyse, les règles de Bioche sont des règles de changement de variable dans le calcul d'intégrales comportant des fonctions trigonométriques. (fr)
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- 6040 (xsd:nonNegativeInteger)
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- En mathématiques, et plus précisément en analyse, les règles de Bioche sont des règles de changement de variable dans le calcul d'intégrales comportant des fonctions trigonométriques. (fr)
- Bioche's rules, formulated by the French mathematician (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines. In the following, is a rational expression in and . In order to calculate , consider the integrand . We consider the behavior of this entire integrand, including the , under translation and reflections of the t axis. The translations and reflections are ones that correspond to the symmetries and periodicities of the basic trigonometric functions. Bioche's rules state that: (en)
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- Bioche's rules (en)
- Règles de Bioche (fr)
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