Integration by Substitution/Primitive

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 $I$.

The primitive of $f$ can be evaluated by:


 * $\ds \int \map f x \rd x = \int \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$

where $x = \map \phi u$.

Also presented as
This can also be seen in the form:


 * $\ds \int \map f x = \int \map \phi u \dfrac {\d x} {\d u} \rd u$

and:


 * $\ds \int \map F {\map \phi x} \rd x = \int \dfrac {\map F u} {\map {f'} x} \rd u$

where $x = \map \phi u$.