Integration by Substitution/Proof Technique

From ProofWiki
Jump to navigation Jump to search

Integration by Substitution: Proof Technique

The usefulness of the technique of Integration by Substitution stems from the fact that it may be possible to choose $\phi$ such that $\map f {\map \phi u} \dfrac \d {\d u} \map \phi u$ (despite its seeming complexity in this context) may be easier to integrate.

If $\phi$ is a trigonometric function, the use of trigonometric identities to simplify the integrand is called integration by trigonometric substitution (or simply trig substitution).

Care must be taken to address the domain and image of $\phi$.

This consideration frequently arises when inverse trigonometric functions are involved.

Also known as

Because the most usual substitution variable used is $u$, this method is often referred to as $u$-substitution in the source works for introductory-level calculus courses.

Some sources refer to this technique as change of variable, but that has a number of different meanings depending upon context.