Question

Integrate the following functions w.r.t. x.

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

Answer 1

$LetI=\int \frac{1}{x^2(x^4+1)}dx=\int \frac{1}{x^2{\left\{x^4(1+\frac{1}{x^4})\right\}}^{3/4}}dx=\int \frac{1}{x^2.x^3{(1+\frac{1}{x^4})}^{3/4}}dx=\int \frac{1}{x^5{(1+\frac{1}{x^4})}^{3/4}}dxPut1+\frac{1}{x^4}=t\Rightarrow -\frac{4}{x^5}dx=dt\Rightarrow \frac{1}{x^5}dx=-\frac{dt}{4}\therefore I=\frac{1}{-4}\int \frac{1}{t^{3/4}}dt=-\frac{1}{4}\left[\frac{t^{1/4}}{1/4}\right]+C=-(1+x^{-4}{)}^{1/4}+C$