# Recursive version of integration by parts

## Contents

## General description of technique

The general procedure is as follows:

- Start doing the integration by parts.
- If necessary, apply integration by parts twice OR use some trigonometric identity with the ultimate goal of seeing the original integration appear
*again*. Note that you should take care to avoid the circular trap. - Denote by the particular choice of antiderivative for which the left and right sides are equal on the nose,i.e., not just up to additive constants.
- Solve the linear equation in .
- The general indefinite integral is this solution plus an arbitrary additive constant.

## Examples

### Sine-squared function

`For further information, refer: Sine-squared function#Integration`

There are many ways of integrating . One of these uses the recursive version of integration by parts. This method is given below:

We now rewrite and obtain:

Setting to be a choice of antiderivative so that the above holds without any freely floating constants, we get:

Rearranging, we get:

This gives:

So the general antiderivative is:

### Secant-cubed function

`For further information, refer: Secant-cubed function#Integration`

We rewrite and perform integration by parts, taking as the part to integrate. We use that an antiderivative of is whereas the derivative of is :

We now use the fact that , or more explicitly, , to rewrite this as:

We now use the integration of the secant function to simplify this as:

We can choose an antiderivative of so that the above equality (between the left-most and right-most expression) holds without any additive constant adjustment, and we get:

We rearrange and obtain:

Dividing by 2, we get:

The general antiderivative expression is thus:

### Exponential times cosine function

Consider the integration problem:

We proceed by taking as the part to integrate and as the part to differentiate. We get:

To avoid the circular trap, we should pick the exponential function as the part to integrate again, getting:

Now, we take as the antiderivative where the above holds without additive constants, and get:

Rearranging, we get:

This gives us:

So the general integral is: