# Linear Change of Variable Rule

How to solve the integral of f(ax + b) by using the linear change of variable rule: formula, 2 examples, and their solutions.

## Formula

### Formula

Let's say that

f is the derivative of F.

Then the derivative of [1/a]⋅F(ax + b) + C is

[1/a]⋅[f(ax + b)⋅a] = f(ax + b).

Derivative of a Composite Function

And integral and derivative

are the opposite operations.

So the integral of f(ax + b) is

[1/a]⋅F(ax + b) + C.

This is the linear change of variable rule.

First write 1/a.

See ax + b as a whole

and write the integral of the outer function:

F(ax + b).

And write the constant term +C.

## Example 1

### Example

Previously, you've solved this example.

Let's solve this example

by using the linear change of variable rule.

### Solution

2x - 1 is the inner linear function.

And [whole]^{8} is the outer function.

The coefficient of the inner linear function is 2.

The reciprocal of 2 is 1/2.

So write 1/2.

See 2x - 1 as a whole.

Then the integral of the outer function [whole]^{8} is

[1/9][whole]^{9}: [1/9](2x - 1)^{9}.

Integral of a Polynomial

The given integral is an indefinite integral.

So write +C.

[1/2]⋅[1/9] = 1/18

So [1/18](2x - 1)^{9} + C is the answer.

## Example 2

### Example

### Solution

x - 3 is the inner linear function.

And 1/[whole] is the outer function.

The coefficient of the inner linear function is 1.

The reciprocal of 1 is 1.

So write 1.

See x - 3 as a whole.

Then the integral of the outer function 1/[whole] is

ln |[whole]|: ln |x - 3|.

Integral of 1/x

The given integral is an indefinite integral.

So write +C.

1⋅ln |x - 3| = ln |x - 3|

So ln |x - 3| + C is the answer.