# How do you integrate (2-x)/(x^2+5x) using partial fractions?

Nov 19, 2016

The answer is $= \frac{2}{5 x} - \frac{7}{5 \left(x + 5\right)}$

#### Explanation:

Let's go to the decomposition in partial fractions

$\frac{2 - x}{{x}^{2} + 5 x} = \frac{2 - x}{\left(x\right) \left(x + 5\right)}$

$= \frac{A}{x} + \frac{B}{x + 5} = \frac{A \left(x + 5\right) + B x}{x \left(x + 5\right)}$

so, $\left(2 - x\right) = A \left(x + 5\right) + B x$

Let $x = 0$, $\implies$$2 = 5 A$ ; $\implies$ , $A = \frac{2}{5}$

Let $x = - 5$ ; $\implies$ , $7 = - 5 B$ ; $\implies$ , $B = - \frac{7}{5}$

$\frac{2 - x}{{x}^{2} + 5 x} = \frac{\frac{2}{5}}{x} + \frac{- \frac{7}{5}}{x + 5} =$