Question

\(\int _{0}^{\pi /4}\log (1+\tan x)dx=\ldots\)

• A. \(\frac{\pi }{4}\log (2)\)
• B. \(\frac{\pi }{2}\log (2)\)
• C. \(\frac{\pi }{8}\log (2)\)
• D. \(\log (2)\)

Hint:

Use the property of definite integrals with x replaced by pi/4 - x and add the two forms.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let \(I=\int_{0}^{\pi/4}\log(1+\tan x)dx\).

Step 2: Apply the Property
Replace \(x\) by \(\dfrac{\pi}{4}-x\): \(1+\tan\left(\dfrac{\pi}{4}-x\right)=1+\dfrac{1-\tan x}{1+\tan x}=\dfrac{2}{1+\tan x}\).

Step 3: Add the Two Forms
\(I=\int_{0}^{\pi/4}\log\dfrac{2}{1+\tan x}dx\).
Adding: \(2I=\int_{0}^{\pi/4}\log 2\,dx=\dfrac{\pi}{4}\log 2\).

Step 4: Solve
\(I=\dfrac{\pi}{8}\log 2\).

Final Answer:
The value of the integral is \(\dfrac{\pi}{8}\log 2\). \[ \boxed{\text{(C) }\dfrac{\pi }{8}\log (2)} \]