Question

Integrate the following functions.
$\int \frac{x^2}{\sqrt{x^6+a^6}}dx.$

Answer 1

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