Integrating with Partial Fractions (Edexcel A Level Further Maths: Core Pure): 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 $(ax+b)$

• With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e. ${(ax+b)}^2$

• In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant $(A,B,C,\mathrm{etc})$

• 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 $(a{x}^2+bx+c)$; very often $b=0$ leaving it as $(a{x}^2+c)$

  • the numerator of the quadratic partial fraction could be of linear form, $(Ax+B)$

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

    • $x=0$ 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 $a{x}^2+c$

  • If it is not then you can get it to look like this by completing the square

• Split into to fraction $\frac{Ax+B}{a{x}^2+c}=\frac{Ax}{a{x}^2+c}+\frac{B}{a{x}^2+c}$

• Integrate $\frac{Ax}{a{x}^2+c}$ using logarithms to get $\frac{A}{2a}\ln \left|a{x}^2+c\right|$

• Integrate $\frac{B}{a{x}^2+c}$ using the formula booklet or using a trigonometric or hyperbolic substitution

  • If a and c have the same sign then use $x=\sqrt{\frac{c}{a}}\tan \left(u\right)$

  • If a and c have different signs then use $x=\sqrt{-\frac{c}{a}}\tanh \left(u\right)$

    • Or in this case you can factorise using surds and then use partial fractions