Integrating f'(x)/f(x) (Cambridge International A-Level Mathematics)

f'(x)/f(x)

Integrating f'(x)/f(x)

How do I integrate 1/x?

• The technique for integrating fractions depends on the type of fraction
• For polynomial denominators see Integration using Partial Fractions
• If $\frac{dy}{dx}=\frac{1}{x}$ then y = ln |x| + c – see Integrating Other Functions
• The type of fraction dealt with here is a specific case of Reverse Chain Rule

How do I integrate f'(x)/f(x)?

• “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

How do I adjust a fraction into the form f'(x)/f(x)?

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

Exam Hint

• 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