Edexcel A Level Further Maths: Core Pure:复习笔记5.2.3 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