# How do you integrate x^3/((x^2+5)^2)?

Mar 26, 2015

$\int {x}^{3} / {\left({x}^{2} + 5\right)}^{2} \mathrm{dx}$

We can rewrite :

${x}^{3} / {\left({x}^{2} + 5\right)}^{2} = \frac{\beta x + \gamma}{{x}^{2} + 5} + \frac{\theta x + \nu}{{x}^{2} + 5} ^ 2$

So : $\frac{\left(\beta x + \gamma\right) \left({x}^{2} + 5\right)}{\left({x}^{2} + 5\right) \left({x}^{2} + 5\right)} + \frac{\theta x + \nu}{{x}^{2} + 5} ^ 2$

Then : $\frac{\beta {x}^{3} + \gamma {x}^{2} + \left(5 \beta + \theta\right) x + \nu + 5 \gamma}{{x}^{2} + 5} ^ 2$

Identification :

$\nu + 5 \gamma = 0$
$5 \beta + \theta = 0$
$\gamma = 0$
$\beta = 1$

So :

$\beta = 1$
$\gamma = 0$
$\theta = - 5$
$\nu = 0$

$\int {x}^{3} / {\left({x}^{2} + 5\right)}^{2} \mathrm{dx} = \int \frac{x}{{x}^{2} + 5} \mathrm{dx} - 5 \int \frac{x}{{x}^{2} + 5} ^ 2 \mathrm{dx}$

$\frac{1}{2} \int \frac{2 x}{{x}^{2} + 5} \mathrm{dx} - \frac{5}{2} \int \frac{2 x}{{x}^{2} + 5} ^ 2 \mathrm{dx}$

$= \left(\frac{1}{2} \ln \left({x}^{2} + 5\right) + \frac{5}{2 {x}^{2} + 10}\right) + C$