# Integration by Parts

## Intuition

Integration by parts will typically apply whenever there are two values of x.

## Procedure

### Basic Example

The task is to integrate the following:

Integration by parts is the product rule in reverse, expressed in simplified form as:

Our first task is to set u, du, v and dv

### Trigonometric Example

The task is to integrate the following:

## Integration by Parts in Python 3

aaa

### With Scipy

kaa

``````import scipy.integrate as integrate
import scipy.special as special
result = integrate.quad(lambda x: special.jv(2.5,x), 0, 4.5)
result``````

## Explanation

Integration by parts applies whenever the integral takes the following form:

The full formula for integration by parts is analagous to the product rule, as follows:

The above equation is often simplified to:

## References

https://www.dummies.com/education/math/calculus/integration-by-parts-problems-where-you-go-around-in-circles/

https://aaronschlegel.me/integration-by-parts.html