7.4 Integration by Partial Fractions

When the integrand is a rational function $\dfrac{P(x)}{Q(x)}$ where $\deg P < \deg Q$ , decompose into partial fractions.

Form of $Q(x)$	Partial Fraction Decomposition
$(x - a)(x - b)$	$\dfrac{A}{x-a} + \dfrac{B}{x-b}$
$(x - a)^2$	$\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$
$(x - a)(x^2 + bx + c)$	$\dfrac{A}{x-a} + \dfrac{Bx + C}{x^2+bx+c}$

Evaluate $\displaystyle\int \frac{1}{(x-1)(x+2)}\,dx$ .

Using partial fractions:

\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}

Solving: $A = \dfrac{1}{3}$ , $B = -\dfrac{1}{3}$ .

\int \frac{1}{(x-1)(x+2)}\,dx = \frac{1}{3}\ln|x-1| - \frac{1}{3}\ln|x+2| + C = \frac{1}{3}\ln\left|\frac{x-1}{x+2}\right| + C