Question

Integrate the following functions.
$\int \frac{x^2}{1-x^6}dx.$

Answer 1

$\int \frac{x^2}{1-x^6}dx$
Let $I=\int \frac{x^2}{1-x^6}dx=\int \frac{x^2}{1-(x^3{)}^2}dx$
Let $x^3=t\Rightarrow 3x^2=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{3x^2}$
$\therefore I=\int \frac{x^2}{1-t^2}\frac{dt}{3x^2}=\frac{1}{3}\int \frac{dt}{1-t^2}=\frac{1}{3}.\frac{1}{2}log\left|\frac{1+t}{1-t}\right|+C[\because \int \frac{dx}{a^2-x^2}=\frac{1}{2a}log\left|\frac{a+x}{a-x}\right|]=\frac{1}{6}log\left|\frac{1+x^3}{1-x^3}\right|+C(\because t=x^3)$