Integrating with Partial Fractions (Edexcel A Level Further Maths): Revision Note

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