Q108 / 180 · KCET 2021 · Mathematics

The value of $\int \frac{x^{2} d x}{\sqrt{x^{6}+a^{6}}}$ is equal to

  • A
    $\log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C$
  • B
    $\log \left|x^{3}-\sqrt{x^{6}+a^{6}}\right|+C$
  • Correct answer
    C
    $\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C$
  • D
    $\frac{1}{3} \log \left|x^{3}-\sqrt{x^{6}+a^{6}}\right|+C$

Solution

Let $I=\int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} d x$ Let $\quad x^{3}=t$ $\begin{aligned} \Rightarrow \quad 3 x^{2} d x &=d t \\ I &=\frac{1}{3} \int \frac{1}{\sqrt{t^{2}+\left(a^{3}\right)^{2}}} d t \\ &=\frac{1}{3} \log \left|t+\sqrt{t^{2}+a^{6}}\right|+C \\ &=\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C \end{aligned}$

Hence, the correct answer is option: 'C'.

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