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 see

 * Weierstrass Substitution