Question

Integrate the following function $w.r.tx$
$\frac{1}{x(x^6+1)}$

Answer 1

$\int \frac{1}{x(x^6+1)}dx$

$=\int \frac{x^5}{x^6(x^6+1)}dx$

$=$ let $x^6=u$

$6x^{5}dx=du$

$=\frac{1}{6}\int \frac{du}{u(u+1)}$

$=\frac{1}{6}\int \frac{1+u-u}{u(u+1)}du$

$=\frac{1}{6}\int \frac{1}{u}-\frac{1}{u+1}du$

$=\frac{1}{6}(inu-in(u+1))$.

$=in\sqrt[6]{6\frac{u}{u+1}}$

$=in\sqrt[6]{6\frac{x^6}{x^6+1}}$

$\therefore =in\frac{x}{\sqrt[6]{6x^6+1}}+c$