AP EAMCET · Maths · Definite Integration
\(\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x=\)
- A \(\pi \log 2+1\)
- B \(\frac{\pi}{2} \log 2+1\)
- C \(\frac{\pi}{4} \log 2\)
- D \(\frac{\pi}{8} \log 2\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{8} \log 2\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & I=\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x \\ & \Rightarrow I=\int_0^{\frac{\pi}{4}} \log \left(1+\tan \left(\frac{\pi}{4}-x\right)\right) d x \\ & \Rightarrow I=\int_0^{\frac{\pi}{4}} \log \left(1+\frac{1-\tan x}{1+\tan x}\right) d x \\ & \Rightarrow…
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