Integrate the function: x 1+2 x^ 2 - Vidyadip

Question

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

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Answer

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

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