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NDA 2017 Maths · Mathematics · Definite Integrals · hard

∫0^π/2ln (tan x) dx equals

Single correct option

∫0π/2ln⁡(tan⁡x) dx\displaystyle\int_0^{\pi/2}\ln(\tan x)\,dx equals
  1. A. π2\frac\pi2
  2. B. ln⁡2\ln2
  3. C. −π2ln⁡2-\frac{\pi}{2}\ln2
  4. D. 00
Show answer and solution

Answer: (D) 00

With x→π2−xx\to\frac\pi2-x, tan⁡x\tan x becomes cot⁡x\cot x, so I=∫0π/2ln⁡(cot⁡x) dx=−II=\int_0^{\pi/2}\ln(\cot x)\,dx=-I. Hence 2I=02I=0 and I=0I=0. The integral converges although ln⁡tan⁡x\ln\tan x is unbounded at both ends. The option −π2ln⁡2-\frac\pi2\ln2 is the value of the related integral ∫0π/2ln⁡sin⁡x dx\int_0^{\pi/2}\ln\sin x\,dx, a common confusion.
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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