AQA A Level Maths: Pure复习笔记8.2.8 Integration by Parts

Integration by Parts

What is Integration by Parts?


  • For integrating the product of two functions - reverse product rule
  • Crucially the product is made from u and dv/dx (rather than u and v)
  • Alternative notation may be used


How do I use Integration by Parts?


  • The hardest part is choosing u and dv/dx as there is no method for doing so
  • u, ideally, becomes simpler when differentiated but this is not always possible
  • dv/dx should be a function that can be integrated fairly easily
  • Be wary of functions that ‘cycle’/’repeat’ when differentiated/integrated
    • ex → ex
    • sin x → cos x → -sin x → -cos x → sin x


  • STEP 1: Choose u and v’, find u’ and v
  • STEP 2: Apply Integration by Parts
    • Simplify anything straightforward
  • STEP 3: Do the ‘second’ integral
    • If an indefinite integral remember “+c”, the constant of integration
  • STEP 4: Simplify and/or apply limits

What happens if I cannot integrate v × du/dx?

  • It is possible integration by parts may need to be applied more than once


ln x


  • A classic ‘set piece’ in almost every A level maths textbook ever written!
  • In general, rewriting f(x) as 1×f(x) can be a powerful problem-solving technique
  • This could be a question in the exam

How do I find a definite integral using parts?


Exam Tip

  • Always think about what an elegant, slick, professional maths solution looks like – solutions normally get more complicated at first but quickly get simpler.
  • If your work is continuing to get more complicated, stop and check for an error.
  • Try to develop a sense of ‘having gone too far down the wrong path’.
  • This general advice is useful to remember:
    • Is the second integral harder than the first?
    • Try swapping your choice of u and dv/dx
    • It is rare to have to apply integration by parts more than twice

Worked Example