# How do you integrate (2x)/(x^2-25) using partial fractions?

Sep 2, 2016

This could be integrated by substitution, but the question specifies partial fractions, so see below.

#### Explanation:

Factor the denominator:

${x}^{2} - 25 = \left(x + 5\right) \left(x - 5\right)$

solve for $A$ and $B$

$\frac{A}{x + 5} + \frac{B}{x - 5} = \frac{2 x}{{x}^{2} - 25}$

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

$A x - 5 A + B x + 5 B = 2 x + 0$

$A + B = 2$
$- 5 A + 5 B = 0$

$A = B = 1$

$\int \frac{2 x}{{x}^{2} - 25} \mathrm{dx} = \int \left(\frac{1}{x + 5} + \frac{1}{x - 5}\right) \mathrm{dx}$

$= \ln \left\mid x + 5 \right\mid + \ln \left\mid x - 5 \right\mid + C$

$= \ln \left\mid {x}^{2} - 25 \right\mid + C$