Q236 / 270 · NDA 2021 (Phase 1) · Mathematics

What is $\int \frac{d x}{\sec ^2\left(\tan ^{-1} x\right)}$ equal to ?

  • A
    $\sin ^{-1} x+c$
  • Correct answer
    B
    $\tan ^{-1} x+c$
  • C
    $\sec ^{-1} x+c$
  • D
    $\cos ^{-1} x+c$

Solution

$\int \frac{d x}{\sec ^2\left(\tan ^{-1} x\right)}$ Let $\tan ^{-1} x=y \Rightarrow \tan y=x$ and $\begin{aligned} & \sec ^2 y=1+\tan ^2 y=1+x^2 \\ & \therefore \int \frac{d x}{\sec ^2\left(\tan ^{-1} x\right)}=\int \frac{d x}{1+x^2}=\tan ^{-1} x+C\end{aligned}$

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

Relevant Questions