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NDA 2020 Maths · Mathematics · Definite Integrals · medium

Using the identity connecting tan ^2x and sec ^2x, the value of ∫0^π/4tan ^2x dx is

Single correct option

Using the identity connecting tan⁡2x\tan^2x and sec⁡2x\sec^2x, the value of ∫0π/4tan⁡2x dx\displaystyle\int_0^{\pi/4}\tan^2x\,dx is
  1. A. 1−π41-\dfrac\pi4
  2. B. π4−1\dfrac\pi4-1
  3. C. 1+π41+\dfrac\pi4
  4. D. π4\dfrac\pi4
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Answer: (A) 1−π41-\dfrac\pi4

Use tan⁡2x=sec⁡2x−1\tan^2x=\sec^2x-1. Then ∫0π/4(sec⁡2x−1) dx=[tan⁡x−x]0π/4=1−π4\displaystyle\int_0^{\pi/4}(\sec^2x-1)\,dx=\left[\tan x-x\right]_0^{\pi/4}=1-\dfrac\pi4. The option π4−1\dfrac\pi4-1 is negative, impossible for a positive integrand. The option π4\dfrac\pi4 would be ∫01dx1+x2\int_0^1\dfrac{dx}{1+x^2} and is a distractor from the inverse tangent integral.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake