Question

Integrate:

$\int x\sqrt{x^2+2}dx$

Answer 1

Consider the given integral.

$I=\int x\sqrt{x^2+2}dx$

Let $t=x^2+2$

$\frac{dt}{dx}=2x+0$

$\frac{dt}{2}=xdx$

Therefore,

$I=\frac{1}{2}\int \sqrt{t}dt$

$I=\frac{1}{2}\left(\frac{t^{3/2}}{\frac{3}{2}}\right)+C$

$I=\frac{t^{3/2}}{3}+C$

On putting the value of $t$, we get

$I=\frac{{(x^2+2)}^{3/2}}{3}+C$

Hence, this is the answer.