AQA A Level Maths: Pure复习笔记8.2.9 Integration using Partial Fractions

Integration using Partial Fractions

What are Partial Fractions?

  • This is the reverse process to adding (or subtracting) fractions
    • The common polynomial denominator is split into factors
  • Make sure you are familiar with the notes on Partial Fractions

8.2.9-Notes-pf_eg

How do I integrate using partial fractions?

  • Fractions with linear denominators can be integrated (See Integrating Other Functions)

8.2.9-Notes-ln_eg

  • A fraction with a polynomial denominator (degree 2+) can be integrated by …
    • … splitting using partial fractions
    • … integrating each partial fraction

8.2.9-Notes-pf_int1 8.2.9-Example-pf_int2

  • STEP 1: Factorise denominator, if needed
  • STEP 2: Express as partial fractions
  • STEP 3: ‘Adjust’ and ‘compensate’ each term so it can be integrated (See Reverse Chain Rule)
  • STEP 4: Integrate each term (usually involves ln) and simplify

Exam Tip

  • When there are several parts, read the whole question:
  • An early part may be a “show that” involving partial fractions
  • A later part may be to integrate the original fraction
  • If you can’t do, or get stuck on, the partial fractions bit of the question you can still use the “show that” result to help with the integration.

Worked Example

8.2.9-Example-soltn_a8.2.9-Example-soltn_b8.2.9-Example-soltn_c

 

 

 

 

 

 

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