Question

Integrate the function : $x\sqrt{1+2x^2}$

Answer 1

Put $1+2x^2=t$
$\Rightarrow 4xdx=dt$
$\Rightarrow \int x\sqrt{1+2x^2}dx=\int \frac{\sqrt{t}dt}{4}$
$=\frac{1}{4}\int t^{\frac{1}{2}}dt$
$=\frac{1}{4}\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$=\frac{1}{6}(1+2x^2{)}^{\frac{3}{2}}+C$