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

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 $\int x\ln xdx$

  • $x$ is "Algebra"

  • $\ln x$ is a Logarithmic function

  • So select $u=\ln x$

    • 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