Integration by Parts (Edexcel A Level Maths) : Revision Note

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Integration by Parts

What is Integration by Parts?

Notes int_parts, AS & A Level Maths revision notes
  • 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

Notes parts_alt_not, AS & A Level Maths revision notes

How do I use Integration by Parts?

Notes parts_choosing, AS & A Level Maths revision notes
  • 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

    • exex

    • sin xcos x-sin x-cos xsin x

    Notes parts_eg_soltn, AS & A Level Maths revision notes

 

  • 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

Notes parts_twice, A Level & AS Level Pure Maths Revision Notes

ln x

Notes int_ln, AS & A Level Maths revision notes
  • 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?

Notes parts_def, A Level & AS Level Pure Maths Revision Notes

Examiner Tips and Tricks

  • 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

Example soltn, AS & A Level Maths revision notes
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Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

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