Explanation

(d)
Put $t = x^{2} + 1 \Rightarrow dt = 2x\ dx$
$\int_{0}^{2}{\frac{x^{3}}{\left( x^{2} + 1 \right)^{3/2}}dx = \frac{1}{2}\int_{1}^{5}{\frac{(t - 1)}{t^{3/2}}dt}}$
$= \frac{1}{2}\int_{1}^{5}{\lbrack t^{- \frac{1}{2}} - t^{- 3/2}}\rbrack dt$
$= \frac{1}{2}\left\lbrack 2\sqrt{t} + 2\frac{1}{\sqrt{t}} \right\rbrack_{1}^{5}$
$= \frac{1}{2}\left\lbrack 2\sqrt{5} + \frac{2}{\sqrt{5}} - 2 - 2 \right\rbrack$
$= \left\lbrack \sqrt{5} + \frac{1}{\sqrt{5}} - 2 \right\rbrack = \frac{6 - 2\sqrt{5}}{\sqrt{5}}$