# How do you express (x^(2) - 1) / (x(x^(2)+1)) in partial fractions?

Nov 26, 2016

The answer is $= - \frac{1}{x} + \frac{2 x}{{x}^{2} + 1}$

#### Explanation:

Let's do the decomposition into partial fractions

$\frac{{x}^{2} - 1}{x \left({x}^{2} + 1\right)} = \frac{A}{x} + \frac{B x + C}{{x}^{2} + 1}$

$= \frac{A \left({x}^{2} + 1\right) + x \left(B x + C\right)}{x \left({x}^{2} + 1\right)}$

Therefore,

${x}^{2} - 1 = A \left({x}^{2} + 1\right) + x \left(B x + C\right)$

Let $x = 0$, $\implies$ $- 1 = A$

Coefficients of ${x}^{2}$

$1 = A + B$, $\implies$, $B = 1 - A = 2$

coefficients of $x$

$0 = C$

Finally, we have

$\frac{{x}^{2} - 1}{x \left({x}^{2} + 1\right)} = - \frac{1}{x} + \frac{2 x}{{x}^{2} + 1}$