Reverse Chain Rule (Cambridge International A-Level Mathematics)

Reverse Chain Rule

How do you integrate (ax + b)ⁿ?

• The reverse chain rule can be used for integrating functions in the form y = (ax + b)ⁿ

  • Make sure you are confident using the chain rule to differentiate functions in the form y = (ax + b)ⁿ
  • The reverse chain rule works backwards

• For n = 2 you will most likely expand the brackets and integrate each term separately

• If n > 2 this becomes time-consuming and if n is not a positive integer we need a different method completely

• To use the reverse chain rule $\int ( a x + b )^{n} d x$(provided n is not -1)

  • Raise the power of n by 1

  • Divide by this new power

  • Divide this whole function by the coefficient of x

    • \int ( a x + b )^{n \mathrm{d} x = \frac{( a x + b )^{n} + 1} }{n + 1} \times \frac{1}{a} + c

• You can check your answer by differentiating it

  • You should get the original function when you differentiate your answer

• Note that this method only works when the function in the brackets is linear (ax + b)

Worked Example

Exam Hint Make sure you can recognise when a question needs to use the reverse chain rule, it may not always be obvious.