Integration by Partial Fractions

A rational function — a ratio of two polynomials — often cannot be integrated directly, but can be split into a sum of simpler fractions that each ARE standard integrals. This splitting is the method of partial fractions.

When it applies

Use partial fractions when the integrand is $\dfrac{P(x)}{Q(x)}$ with the degree of $P$ less than the degree of $Q$ (a proper fraction), and $Q(x)$ factorises into linear (or repeated/quadratic) factors. If the fraction is improper (degree of $P \ge$ degree of $Q$), first divide to get a polynomial plus a proper fraction.

The standard decomposition (distinct linear factors)

For a denominator that factorises into distinct linear factors, write one partial fraction per factor with an unknown constant on top:

Multiply through by the denominator to clear fractions, then find $A$ and $B$ — most quickly by substituting the values $x=\alpha$ and $x=\beta$ that make one bracket vanish. Each resulting fraction integrates by the log rule, $\displaystyle\int\frac{dx}{x-\alpha} = \log\lvert x-\alpha\rvert + c$.