# Linear Change of Variable Rule

How to solve integral problems by using the linear change of variable rule: formula, proof, examples, and their solutions.

## Formula

See the given integral: ∫ *f*(*ax* + *b*) *dx*.

The linear part is *ax* + *b*.

Then the given integral is:

write the reciprocal of *a*, 1/*a*,

write the antiderivative, *F*(*ax* + *b*),

and write + *C*.

## Proof

Set *ax* + *b* = *t*.

Then *a* *dx* = *dt*,*dx* = (1/*a*) *dt*.

Substitute [*ax* + *b*] with [*t*].

And substitute [*dx*] with [(1/*a*) *dt*].

Then (given) = ∫ *f*(*t*) (1/*a*) *dt*.

Integration by substitution (Part 1)

Take 1/*a* out from the integral.

Find the indefinite integral.

Put *ax* + *b* in *t*.

Then (given) = (1/*a*)⋅*F*(*ax* + *b*) + *C*.

So ∫ *f*(*ax* + *b*) *dx* = (1/*a*)⋅*F*(*ax* + *b*) + *C*.

## Example 1

Previously, you've solved this example.

Integration by substitution (Part 1)

Let's solve the same example

by using the linear change of variable rule.

The linear part is 2*x* - 1.

So write the reciprocal of 2, 1/2,

write the antiderivative, (1/9)(2*x* - 1)^{9},

and write +*C*.

So (given) = (1/18)(2*x* - 1)^{9} + *C*.

As you can see,

you can get the same answer,

much easier than the previous solution.

This is the reason

why the linear change of variable rule is used.

## Example 2

The linear part is *x* - 3.

So write the reciprocal of 1, 1,

write the antiderivative, ln |*x* - 3|,

and write +*C*.

So (given) = ln |*x* - 3| + *C*.

Indefinite integration of 1/*x*

Just like this example,

the linear change of variable rule

is also frequently used

when integrating fractions,

which you'll see in the next page.