Integrating with Partial Fractions (Edexcel A Level Further Maths: Core Pure)

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Integrating with Partial Fractions

What is meant by partial fractions with quadratic denominators?

  • For linear denominators the denominator of the original fraction can be factorised such that the denominator becomes a product of linear terms of the form left parenthesis a x space plus space b right parenthesis
  • With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e. left parenthesis a x space plus space b right parenthesis squared
  • In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant left parenthesis A comma space B comma space C comma space etc right parenthesis
  • For this course, quadratic denominators refer to fractions that contain a quadratic factor (that cannot be factorised) on the denominator
    • the denominator of the quadratic partial fraction will be of the form left parenthesis a x squared space plus space b x space plus space c right parenthesis; very often b space equals space 0 leaving it as left parenthesis a x squared space plus space c right parenthesis
    • the numerator of the quadratic partial fraction could be of linear form, left parenthesis A x space plus space B right parenthesis

How do I find partial fractions involving quadratic denominators?

  •  STEP 1          Factorise the denominator as far as possible (if not already done so)
    • Sometimes the numerator can be factorised too
  • STEP 2          Split the fraction into a sum with
    • the linear denominator having an (unknown) constant numerator
    • the quadratic denominator having an (unknown) linear numerator
  • STEP 3          Multiply through by the denominator to eliminate fractions
  • STEP 4          Substitute values into the identity and solve for the unknown constants
    • Use the root of the linear factor as a value of x to find one of the unknowns
    • Use any two values for x to form two equations to solve simultaneously
      • x space equals space 0 is a good choice if this has not already been used with the linear factor
  • STEP 5          Write the original as partial fraction

How do I integrate the fraction with the quadratic denominator?

  •  The quadratic denominator will be of the form a x squared plus c
    • If it is not then you can get it to look like this by completing the square
  • Split into to fraction fraction numerator A x plus B over denominator a x squared plus c end fraction equals fraction numerator A x over denominator a x squared plus c end fraction plus fraction numerator B over denominator a x squared plus c end fraction
  • Integrate fraction numerator A x over denominator a x squared plus c end fraction using logarithms to get fraction numerator A over denominator 2 a end fraction ln open vertical bar a x squared plus c close vertical bar
  • Integrate fraction numerator B over denominator a x squared plus c end fraction using the formula booklet or using a trigonometric or hyperbolic substitution
    • If a and have the same sign then use x equals square root of c over a end root tan open parentheses u close parentheses
    • If a and have different signs then use x equals square root of negative c over a end root tanh open parentheses u close parentheses
      • Or in this case you can factorise using surds and then use partial fractions

Worked example

Find integral fraction numerator 8 x squared minus 9 x over denominator left parenthesis x minus 3 right parenthesis left parenthesis 4 x squared plus 9 right parenthesis end fraction d x

5-2-3-edex-fm--alevel-we1-hypsub-soltn-a1

5-2-3-edex-fm--alevel-we1-hypsub-soltn-a2

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Paul

Author: Paul

Expertise: Maths

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 – one of the many reasons he is excited to be a member of the SME team.