5.2.3 Integrating with Partial Fractions | Edexcel A Level Further Maths: Core Pure Revision Notes 2017

Integrating with Partial Fractions

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 $(a x + b)$
With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e. ${(a x + b)}^{2}$
In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant $(A, B, C, etc)$
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 $(a x^{2} + b x + c)$ ; very often $b = 0$ leaving it as $(a x^{2} + c)$
- the numerator of the quadratic partial fraction could be of linear form, $(A x + B)$

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

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 $\frac{A x + B}{a x^{2} + c} = \frac{A x}{a x^{2} + c} + \frac{B}{a x^{2} + c}$
Integrate $\frac{A x}{a x^{2} + c}$ using logarithms to get $\frac{A}{2 a} \ln |a x^{2} + c|$
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 $x = \sqrt{\frac{c}{a}} \tan (u)$
- If a and c have different signs then use $x = \sqrt{- \frac{c}{a}} \tanh (u)$
  - Or in this case you can factorise using surds and then use partial fractions

Find $\int \frac{8 x^{2} - 9 x}{(x - 3) (4 x^{2} + 9)} d x$

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