Integration by Substitution

Theorem
Let $\phi$ be a real function which has a derivative on the closed interval $\closedint a b$.

Let $I$ be an open interval which contains the image of $\closedint a b$ under $\phi$.

Let $f$ be a real function which is continuous on $I$.

Definite Integral
The technique of solving an integral in this manner is called integration by substitution.

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.

Also see

 * Weierstrass Substitution