Linear Change of Variable Rule

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

(integral of f(ax + b) dx) = (1/a)*F(ax + b) + C

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

Linear Change of Variable Rule: Proof of the Formula

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

Find the given indefinite integral. The integral of (2x - 1)^8 dx

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 2x - 1.

So write the reciprocal of 2, 1/2,
write the antiderivative, (1/9)(2x - 1)9,
and write +C.

So (given) = (1/18)(2x - 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

Find the given indefinite integral. The integral of [1 / (x - 3)] dx

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.