The Reverse Chain Rule
The reverse chain rule integrates composite functions of the form \(\displaystyle\int f'(x)\,g'\!\big(f(x)\big)\,dx\), recognising a function and its derivative together.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
The reverse chain rule integrates a bracket raised to a power when the inside's derivative is present. This Year 12 Mathematics Advanced topic (MAV-12-05) undoes the chain rule.
\(\displaystyle\int (ax+b)^{n}dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\): integrate as a power, then divide by the inner coefficient \(a\).
More generally \(\displaystyle\int f'(x)[f(x)]^{n}dx=\dfrac{[f(x)]^{n+1}}{n+1}+C\) when the derivative of the inside appears as a factor.
Method
- Identify the inside and its derivative.
- Raise the power by one.
- Divide by the new power and by the inner coefficient.
| \(\ \) | \(=\) | \(\dfrac{(2x+1)^{4}}{4\times2}+C\) |
| \(=\) | \(\dfrac{(2x+1)^{4}}{8}+C\) |
| \(\ \) | \(=\) | \(\dfrac{(x^{2}+1)^{5}}{5}+C\) |
| \(\ \) | \(=\) | \(\dfrac{(3x-2)^{6}}{18}+C\) |
| \(\ \) | \(=\) | \(\dfrac{(5x+1)^{3}}{15}+C\) |
Common pitfalls
Frequently asked questions
What is the reverse chain rule?
It is a way to integrate a function raised to a power by undoing the chain rule: raise the power by one and divide by the new power and the inner coefficient.
How do you integrate (2x+1) cubed?
Raise the power to 4 and divide by 4 and by the inner coefficient 2, giving (2x+1) to the fourth over 8, plus C.
When can you use the reverse chain rule directly?
When the derivative of the inside function appears as a factor, or when the inside is linear so its derivative is just a constant.
How do you check a reverse chain rule integral?
Differentiate your answer with the chain rule; it should return the original integrand.