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The value of $\int \frac{x^{2} d x}{\sqrt{x^{6}+a^{6}}}$ is equal to
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Verified Answer
The correct answer is:
$\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C$
Let $I=\int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} d x$
Let $\quad x^{3}=t$
$\begin{aligned}
\Rightarrow \quad 3 x^{2} d x &=d t \\
I &=\frac{1}{3} \int \frac{1}{\sqrt{t^{2}+\left(a^{3}\right)^{2}}} d t \\
&=\frac{1}{3} \log \left|t+\sqrt{t^{2}+a^{6}}\right|+C \\
&=\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C
\end{aligned}$
Let $\quad x^{3}=t$
$\begin{aligned}
\Rightarrow \quad 3 x^{2} d x &=d t \\
I &=\frac{1}{3} \int \frac{1}{\sqrt{t^{2}+\left(a^{3}\right)^{2}}} d t \\
&=\frac{1}{3} \log \left|t+\sqrt{t^{2}+a^{6}}\right|+C \\
&=\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C
\end{aligned}$
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