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Integration by Partial Fractions

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Question

Solve :
$\int \frac{(x - 1)}{(x + 1) (x - 2)} d x$

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Solution

$\begin{matrix} \int \frac{(x - 1)}{(x + 1) (x - 2)} d x L e t \frac{(x - 1)}{(x + 1) (x - 2)} = \frac{A}{(x + 1)} + \frac{B}{(x - 2)} T h e n, (x - 1) \equiv A (x - 2) + B (x + 1) . . . . . . (i) p u t t i n g x = 1 i n (i), w e g e t A = \frac{2}{3} . p u t t i n g x = 2 i n (i), w e g e t B = \frac{1}{3} . ∴ \frac{(x - 1)}{(x + 1) (x - 2)} = \frac{2}{3 (x + 1)} + \frac{1}{3 (x - 2)} = \int \frac{(x - 1)}{(x + 1) (x - 2)} d x = \frac{2}{3} \int \frac{d x}{(x + 1)} + \int \frac{d x}{(x - 2)} = \frac{2}{3} log | x + 1 | + \frac{1}{3} log | x - 2 | + C \end{matrix}$

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