Reverse Chain Rule (Cambridge (CIE) A Level Maths) : Revision Note

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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 integral left parenthesis a x plus b right parenthesis to the power of 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

      • integral left parenthesis a x blank plus blank b right parenthesis to the power of n blank straight d x equals fraction numerator open parentheses a x plus b close parentheses to the power of n plus 1 end exponent over denominator n plus 1 end fraction cross times 1 over a plus 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

6-1-4-reverse-chain-rule-we-solution

Examiner Tips and Tricks

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

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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.