Question

Evaluate $\int \frac{x^2}{\sqrt{x^6+a^6}}dx$

Answer 1

$\int \frac{x^2}{\sqrt{x^6+a^6}}dx$

$\int \frac{x^2}{\sqrt{{\left(x\right)}^2+{\left(a^3\right)}^2}}dx$

Putting $x^3=t$

$3x^{2}dx=dt$

$\Rightarrow x^{2}dx=\frac{dt}{3}$

$\frac{1}{3}\int \frac{dt}{\sqrt{t^2+{\left(a^3\right)}^2}}$

$\frac{1}{3}\left[\log \left|t+\sqrt{t^2+a^6}\right|\right]+c$

$\frac{1}{3}\left[\log \left|x^3\sqrt{x^6+a^6}\right|\right]+c$