Integration by Substitution (Edexcel IAL Maths)

Substitution (Reverse Chain Rule)

Spec point — 6.2

What is integration by substitution?

• 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’

How do I integrate when a substitution is not given?

• 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

Exam Hint

• 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.

Worked Example