CIE A Level Maths: Pure 3复习笔记5.1.4 Substitution (Reverse Chain Rule)

Substitution (Reverse Chain Rule)

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’8.2.4-Notes-subst_eq

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 inor
    • … evaluate the definite integral using the u limits
  • STEP 5: Find c, the constant of integration, if needed


8.2.4-Notes-subst_eg2_def1 8.2.4-Notes-subst_eg2_def2

Exam Tip

  • 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