# Category:Integration by Substitution

This category contains pages concerning **Integration by Substitution**:

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 $I$.

### Primitive

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 $x = \map \phi u$.

### Definite Integral

If $\map \phi a \le \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:

- $\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$

where $t = \map \phi u$.

If $\map \phi a > \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:

- $\ds - \int_{\map \phi b}^{\map \phi a} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$

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

## Subcategories

This category has only the following subcategory.

### I

## Pages in category "Integration by Substitution"

The following 8 pages are in this category, out of 8 total.

### I

- Integration by Substitution
- Integration by Substitution/Also known as
- Integration by Substitution/Corollary
- Integration by Substitution/Definite Integral
- Integration by Substitution/Primitive
- Integration by Substitution/Primitive/Proof 1
- Integration by Substitution/Primitive/Proof 2
- Integration by Substitution/Proof Technique