Question:

$I = \int \frac{x^3}{\sqrt{x^2+2}} dx = \int \frac{x^2 (xdx)}{\sqrt{x^2+2}}$<br>Put $x^2 + 2 = t^2 \Rightarrow 2xdx = 2tdt \Rightarrow xdx = tdt$<br>$\therefore I = \int \frac{(t^2 - 2) \cdot (tdt)}{t} = \int (t^2 - 2) dt$<br>$= \frac{t^3}{3} - 2t + C = \frac{(x^2 + 2)^{\frac{3}{2}}}{3} - 2\sqrt{x^2 + 2} + C$