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.