$\int x^{2} \sqrt{x^{6} + 1} is equal to$

Moderate

A

$\frac{1}{3} [\frac{x^{3}}{3} \sqrt{x^{6} + 1} + \frac{1}{2} \log |x^{3} + \sqrt{x^{6} + 1}|] + C$
B

$\frac{1}{3} [\frac{x^{3}}{2} \sqrt{x^{6} + 1} - \frac{1}{2} \log |x^{3} + \sqrt{x^{6} + 1}|] + C$
C

$\frac{1}{3} [\frac{x^{3}}{2} \sqrt{x^{6} + 1} + \frac{1}{2} \log |x + \sqrt{x + 1}|] + C$
D

$\frac{1}{3} [\frac{x^{3}}{2} \sqrt{x^{6} + 1} + \frac{1}{2} \log |x^{3} + \sqrt{x^{6} + 1}|] + C$

Solution

$Let I = \int x^{2} \sqrt{x^{6} + 1} d x$
$= \int x^{2} \sqrt{{(x^{3})}^{2} + 1} d x$
$Put x^{3} = t$
$\begin{array}{l} x^{2} d x = \frac{d t}{3} \\ I = \frac{1}{3} \int \sqrt{t^{2} + 1} d x \\ I = \frac{1}{3} [\frac{t}{2} \sqrt{t^{2} + 1} + \frac{1}{2} \log |t + \sqrt{t^{2} + 1}|] + C \\ = \frac{1}{3} [\frac{x^{3}}{2} \sqrt{x^{6} + 1} + \frac{1}{2} \log |x^{3} + \sqrt{x^{6} + 1}|] + C \end{array}$

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