AQA A Level Maths: Pure复习笔记8.2.4 f'(x)/f(x)

f'(x)/f(x)

Integrating fractions

  • The technique for integrating fractions depends on the type of fraction
  • For polynomial denominators see Integration using Partial Fractions
  • The type of fraction dealt with here is a specific case of Reverse Chain Rule

8.2.7-Notes-diff_lnfx

8.2.7-Notes-top_diff_bot

  • “The top is ‘almost’ the derivative of the bottom”
    • 'almost' here meaning 'a multiple of' (see below)
  • The integral will involve ln |f(x)| - ie ln of the bottom
    • Due to reverse chain rule

8.2.7-Notes-f%E2%80%99_f_eg

Why ‘almost’?

8.2.7-Notes-f%E2%80%99_f_eg_adj_comp

  • There may be coefficients to ‘adjust’ and ‘compensate’ for

Exam Tip

  • If you're unsure if the fraction is of the form f’(x)/f(x), differentiate the denominator.
  • Compare this to the numerator but you can ignore any coefficients.
  • If the coefficients do not match then ‘adjust’ and ‘compensate’ for them.

Worked Example

8.2.7-Example-soltn18.2.7-Example-soltn2

 

 

 

 

 

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