Integrating after partial fractions
Decomposition is the algebra stage. Integration begins only after the rational function has been split into standard pieces.
Complete the algebra before integrating
A rational function is one polynomial divided by another. The degree of a polynomial is its highest power, so has degree 2. A rational function is proper when the numerator degree is lower than the denominator degree. If it is improper, divide the polynomials first.
- A factor needs both and .
- An irreducible quadratic such as needs a linear numerator . Irreducible means it cannot be factorised into real linear factors, so the quadratic stays as one denominator.
- Check the decomposition by recombining the fractions before integrating.
Examiner note
Classify every piece after decomposition
Each piece has its own ending
A squared repeated factor is not a logarithm.
- A linear denominator gives a logarithm, with its linear coefficient divided out.
- A repeated squared factor follows the power rule and gives a reciprocal, not another logarithm.
- Over , split the linear numerator into a derivative part and a constant part.
Key idea
Handle every repeated-factor term
Worked example
Use the decomposition from Further Algebra
Integrate the three pieces separately:
Common mistake
(Pure Mathematics 2 & 3 Coursebook, Ch. 8.5)
Finish a quadratic factor in two parts
From the decomposition:
The -piece gives a logarithm; the constant -piece gives an inverse tangent.
Hence integrate , using the decomposition shown above.