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NDA 2019 Maths · Mathematics · Integrals · medium

A student simplifies the denominator of (1)/(1+cos x) with a half-angle identity before integrating it with respect to x. The antiderivative obtained…

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A student simplifies the denominator of 11+cos⁡x\dfrac{1}{1+\cos x} with a half-angle identity before integrating it with respect to xx. The antiderivative obtained is
  1. A. −cot⁡x2+C-\cot\frac x2+C
  2. B. ln⁡(1+cos⁡x)+C\ln(1+\cos x)+C
  3. C. tan⁡x+C\tan x+C
  4. D. tan⁡x2+C\tan\frac x2+C
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Answer: (D) tan⁡x2+C\tan\frac x2+C

Use 1+cos⁡x=2cos⁡2x21+\cos x=2\cos^2\frac x2, so the integrand is 12sec⁡2x2\frac12\sec^2\frac x2. Integrating, 12⋅2tan⁡x2+C=tan⁡x2+C\frac12\cdot2\tan\frac x2+C=\tan\frac x2+C. The option −cot⁡x2-\cot\frac x2 is the antiderivative of 11−cos⁡x\frac{1}{1-\cos x}, and tan⁡x\tan x ignores the half-angle.
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Original question written in the NDA 2019 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