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

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Solution

$\frac{x}{(x - 1) (x^{2} + 1)} = \frac{A}{x - 1} + \frac{B x + C}{x^{2} + 1}$
$⟹ x = A (x^{2} + 1) + (B x + C) (x - 1)$
$⟹ A + B = 0 ⟹ B = - A, A - C = 0 ⟹ C = A, C - B = 1 ⟹ 2 A = 1 ⟹ A = - B = C = \frac{1}{2}$
$\int \frac{x d x}{(x - 1) (x^{2} + 1)} = \frac{1}{2} \int \frac{d x}{x - 1} - \frac{1}{4} \int \frac{2 x}{x^{2} + 1} d x + \frac{1}{2} \int \frac{d x}{x^{2} + 1} = \frac{1}{2} ln (x - 1) - \frac{1}{4} ln (x^{2} + 1) + \frac{1}{2} {tan}^{- 1} x + C$

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