# How do you express (3x )/ (x^2 * (x^2+1) ) in partial fractions?

May 8, 2016

$\frac{3 x}{{x}^{2} \cdot \left({x}^{2} + 1\right)} = \frac{3}{x} - \frac{3 x}{{x}^{2} + 1}$

#### Explanation:

(3x)/(x^2*(x^2+1))=3/(x*(x^2+1) and let its partial fractions be

$\frac{3}{x \cdot \left({x}^{2} + 1\right)} \Leftrightarrow \frac{A}{x} + \frac{B x + C}{{x}^{2} + 1}$ or

$\frac{3}{x \cdot \left({x}^{2} + 1\right)} \Leftrightarrow \frac{A \left({x}^{2} + 1\right) + x \left(B x + C\right)}{x \cdot \left({x}^{2} + 1\right)}$ or

$\frac{3}{x \cdot \left({x}^{2} + 1\right)} \Leftrightarrow \frac{{x}^{2} \left(A + B\right) + C x + A}{x \cdot \left({x}^{2} + 1\right)}$ or

$A + B = 0$, $C = 0$ and $A = 3$. Thus $B = - 3$

Hence $\frac{3 x}{{x}^{2} \cdot \left({x}^{2} + 1\right)} = \frac{3}{x} - \frac{3 x}{{x}^{2} + 1}$