The value of integral \(∈t 0 π / 2 log tan x d x\) is

\(I=∈t 0 π / 2 log tan x d x\) \[\] (i) Using property \(∈t 0 a f(x) d x=∈t 0 a f(a-x) d x\) \[\] \(I=∈t 0 π / 2 log tan (π/2-x) d x\) \[\] \(I=∈t 0 π / 2 log cot x d x\) \[\] Adding (i) & (ii) \[\] \(2 I=0 I=0\) \[\] Choice (D)\[\]

The value of integral \(∈t 0 π / 2 log... - Solved

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Question 14

Mathematics Definite Integrals Medium

The value of integral \(\int_{0}^{\pi / 2} \log \tan x d x\) is

(A) \(\pi\)

(B) \(\frac{\pi}{2}\)

(C) \(\frac{\pi}{3}\)

(D) 0

✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\(I=\int_{0}^{\pi / 2} \log \tan x d x\) \[\] (i) Using property \(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\) \[\] \(I=\int_{0}^{\pi / 2} \log \tan \left(\frac{\pi}{2}-x\right) d x\) \[\] \(I=\int_{0}^{\pi / 2} \log \cot x d x\) \[\] Adding (i) & (ii) \[\] \(2 I=0 \Rightarrow I=0\) \[\] Choice (D)\[\]

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