Q217 / 270 · NDA 2013 (Phase 2) · Mathematics

What is $\int \frac{d x}{\sqrt{4+x^2}}$ equal to? where c is an arbitrary constant.

  • Correct answer
    A
    $\ln \left|\sqrt{4+x^2}+x\right|+c$
  • B
    $\ln \left|\sqrt{4+x^2}-x\right|+c$
  • C
    $\sin ^{-1}\left(\frac{x}{2}\right)+c$
  • D
    None of the above

Solution

$\begin{aligned} & \mathrm{I}=\int \frac{d x}{\sqrt{x^2+4}}=\int \frac{d x}{\sqrt{x^2+(2)^2}} \\ & \text { Let } \quad x=2 \tan \theta \\ & d x=2 \sec ^2 \theta d \theta \\ & \int \frac{2 \sec ^2 \theta \cdot d \theta}{\sqrt{2^2\left(1+\tan ^2 \theta\right)}}=\int \sec \theta \cdot d \theta \\ & =\ln (\sec \theta+\tan \theta)+C=\ln \left(x+\sqrt{x^2+4}\right)+C \end{aligned}$

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

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