Simplify the fraction before choosing a method. Polynomial division produces a polynomial plus a proper rational fraction.
| Denominator or numerator structure | Method |
|---|---|
| Numerator degree at least denominator degree | Divide first, then inspect the remainder. |
| At most two distinct linear denominator factors | Decompose the proper fraction, then integrate logarithmic terms. |
| A positive quadratic denominator | Complete the square and scale to an inverse-tangent form. |
| A square root of a positive difference of squares | Complete the square and scale to an inverse-sine form. |
| A linear numerator over a quadratic | Express the numerator as a multiple of the denominator derivative plus a constant. |
Partial fractions in AA HL
AHL1.11 specifies a proper fraction with at most two distinct linear denominator factors. AHL5.15 applies that decomposition to integration.
\(\begin{aligned}\dfrac{3x+5}{(x+1)(x+2)}&=\dfrac{2}{x+1}+\dfrac{1}{x+2}\\ \int\dfrac{3x+5}{(x+1)(x+2)}\,\mathrm{d}x&=2\ln|x+1|+\ln|x+2|+C\end{aligned}\)
Keep absolute values in logarithms unless the domain establishes positivity. A definite integral cannot be evaluated across a denominator zero by simply substituting endpoints.
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