Integrate the function x^ 2 1-x^ 6 . - Vidyadip

Question

Integrate the function $\frac{x^{2}}{1-x^{6}}$.

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Answer

Let $x^3 = t$
$\Rightarrow 3x^2 dx = dt$
$\Rightarrow \int \frac{x^{2}}{1-x^{6}} d x=\frac{1}{3} \int \frac{d t}{1-t^{2}}$
$\Rightarrow\frac{1}{3}\left[\frac{1}{2} \log \left|\frac{1+t}{1-t}\right|\right]+C$
$\Rightarrow \frac{1}{6}\left[\log \left|\frac{1+x^{3}}{1-x^{3}}\right|\right]+C$

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