# Please show the integration of (x^3)/((x^4+7)^8)dx step by step?

May 23, 2018

We have

$I = \int {x}^{3} / {\left({x}^{4} + 7\right)}^{8} \mathrm{dx}$

Let $u = {x}^{4} + 7 \mathrm{dx}$. Then $\mathrm{du} = 4 {x}^{3} \mathrm{dx}$ and $\mathrm{dx} = \frac{\mathrm{du}}{4 {x}^{3}}$

$I = \int {x}^{3} / {u}^{8} \cdot \frac{\mathrm{du}}{4 {x}^{3}}$

$I = \frac{1}{4} \int {u}^{-} 8 \mathrm{du}$

$I = \frac{1}{4} \left(- \frac{1}{7} {u}^{-} 7\right) + C$

$I = - \frac{1}{28} {\left({x}^{4} + 7\right)}^{-} 7 + C$

Hopefully this helps!