Integration by Parts (Cambridge International A-Level Mathematics)

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

• 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

  • e^x → e^x
  • 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

How do I choose u for integration by parts?

• A helpful mnemonic is LIATE

• This refers to the order of preference for selecting u

  • L = Logarithms
  • I = Inverse functions
  • A = Algebra (e.g. polynomials)
  • T = Trigonometric functions
  • E = Exponentials

• For example when using integration by parts for integral x ln x d x

  • $x$ is "Algebra"

  • $\mathrm{ln}x$ is a Logarithmic function

  • So select $u=\mathrm{ln}x$

    • And $\frac{dv}{dx}=x$

Can I use integration by parts twice?

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

How do I integrate 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

Can I find definite integrals using integration by parts?

• You can find the value of a definite integral using integration by parts

  • Use the layout shown in the example below

Exam Hint

• 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