Substitution (Reverse Chain Rule)

Make sure you are familiar with Chain Rule and Reverse Chain Rule
In more awkward problems it is harder to spot the reverse chain rule
It is possible to use a substitution
These kinds of substitutions usually will not be given
- The substitution is deemed ‘obvious’

Look to substitute the ‘second’ (rather than the ‘main’) function
STEP 1: Determine the substitution
- What is the ‘main’ function? ‘second’ function?
STEP 2: Differentiate the substitution and rearrange
- du/dx here can be treated like a fraction (eg × dx to get rid of fractions)
STEP 3: Replace all parts of the integral
- all x terms should be replaced with equivalent u terms, including dx
- if a definite integral change the limits from x to u too
STEP 4: Integrate and either …
- …substitute x back in
  or
- … evaluate the definite integral using the u limits
STEP 5: Find c, the constant of integration, if needed

Notes subst_eg_soln2, AS & A Level Maths revision notes

Notes subst_eg2_def1, AS & A Level Maths revision notes Notes subst_eg2_def2, AS & A Level Maths revision notes

A lot of the “work” in these problems happens in your head :
- what is the ‘main’ function?
- what should the substitution be?
- are there any numbers to ‘adjust’ and ‘compensate’ for?
Be sure that what you write down is clear and easy to follow, and remember that you can check your final answer by differentiating it.

Example soltn1, AS & A Level Maths revision notes Example soltn2, AS & A Level Maths revision notes

Substitution (Reverse Chain Rule) (Edexcel International A Level Maths: Pure 4)