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.