Q116 / 180 · COMEDK 2016 · Mathematics

$\int \frac{d x}{x \sqrt{x^{6}-16}}$

  • A
    $\sec ^{-1}\left(\frac{x^{3}}{4}\right)+c$
  • Correct answer
    B
    $\frac{1}{12} \sec ^{-1}\left(\frac{x^{3}}{4}\right)+c$
  • C
    $\cosh ^{-1}\left(\frac{x^{3}}{4}\right)+c$
  • D
    $\frac{1}{3} \sec ^{-1}\left(\frac{x^{3}}{4}\right)+c$

Solution

Let $I=\int \frac{d x}{x \sqrt{\left(x^{3}\right)^{2}-16}}$ Put $x^{3}=t \Rightarrow 3 x^{2} d x=d t$ So, $I=\frac{1}{3} \int \frac{d t}{x^{3} \sqrt{\left(x^{3}\right)^{2}-16}}$ $\begin{aligned} &=\frac{1}{3} \int \frac{d t}{t\left(\sqrt{t^{2}-16}\right.} \\ &=\frac{1}{3 \times 4} \sec ^{-1}\left(\frac{t}{4}\right)+c \\ &=\frac{1}{12} \sec ^{-1}\left(\frac{x^{3}}{4}\right)+c \end{aligned}$

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

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