Edexcel A Level Further Maths: Core Pure:复习笔记5.2.3 Integrating with Partial Fractions

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
  • With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e.
  • In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant
  • 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 ; very often  leaving it as
    • the numerator of the quadratic partial fraction could be of linear form,

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  to find one of the unknowns
    • Use any two values for  to form two equations to solve simultaneously
      • 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
    • If it is not then you can get it to look like this by completing the square
  • Split into to fraction
  • Integrate  using logarithms to get
  • Integrate  using the formula booklet or using a trigonometric or hyperbolic substitution
    • If a and c have the same sign then use
    • If a and c have different signs then use
      • Or in this case you can factorise using surds and then use partial fractions

Worked Example