Question

Evaluate
$\int \frac{x^7}{(1+x^4{)}^2}dx$

Answer 1

$\int \frac{x^{7}dx}{{(1+x^4)}^2}$

Let $u=1+x^4\Rightarrow du=4x^3$

$=\int \frac{x^4\times x^{3}dx}{{(1+x^4)}^2}$

$=\frac{1}{4}\int \frac{(u-1)du}{u^2}$

$=\frac{1}{4}\int \frac{udu}{u^2}-\frac{1}{4}\int \frac{du}{u^2}$

$=\frac{1}{4}\ln \left|u\right|-\frac{1}{4}\left(\frac{-1}{u}\right)+c$

$=\frac{1}{4}[\ln |1+x^4|+\frac{1}{1+x^4}]+c$

where $u=1+x^4$