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NDA 2017 Maths · Mathematics · Definite Integrals · medium

Examine these two claims about definite integrals: 1. ∫0^1(dx)/(√(1-x^2)=(π)/(2). 2. ∫0^π/4sec ^2x dx=(π)/(4).

Single correct option

Examine these two claims about definite integrals:
1. ∫01dx1−x2=π2\displaystyle\int_0^1\frac{dx}{\sqrt{1-x^2}}=\frac\pi2.
2. ∫0π/4sec⁡2x dx=π4\displaystyle\int_0^{\pi/4}\sec^2x\,dx=\frac\pi4.
Which of the statements given above is/are correct?
  1. A. 11 only
  2. B. 22 only
  3. C. Both 11 and 22
  4. D. Neither 11 nor 22
Show answer and solution

Answer: (A) 11 only

1: the antiderivative is sin⁡−1x\sin^{-1}x, so the value is sin⁡−11−sin⁡−10=π2\sin^{-1}1-\sin^{-1}0=\frac\pi2. 2: ∫sec⁡2x dx=tan⁡x\int\sec^2x\,dx=\tan x, so the value is tan⁡π4−tan⁡0=1\tan\frac\pi4-\tan0=1, not π4\frac\pi4; the number π4\frac\pi4 is just the upper limit. Only claim 1 is correct.
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Original question written in the NDA 2017 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