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  4. The value of displaystyle ∫ 0(π /3)log(1 + √3 tan x)dx is equal to

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Q. The value of $\displaystyle \int _{0}^{\frac{\pi }{3}}log\left(1 + \sqrt{3} tan x\right)dx$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

Let, $I =\int_{0}^{\frac{\pi}{3}} \log (1+\sqrt{3} \tan x) d x$
$ \begin{array}{l} I =\int_{0}^{\frac{\pi}{3}} \log \left[1+\sqrt{3} \tan \left(\frac{\pi}{3}-x\right)\right] d x \\ =\int_{0}^{\frac{\pi}{3}}\left[\log \left(1+\sqrt{3} \tan \left(\frac{\sqrt{3}-\tan x}{1+\sqrt{3} \tan x}\right)\right] d x\right. \\ =\int_{0}^{\frac{\pi}{3}} \log \left(\frac{1+\sqrt{3} \tan x+3-\sqrt{3} \tan x}{1+\sqrt{3} \tan x}\right) d x \\ \Rightarrow I=\int_{0}^{\frac{\pi}{3}}(\log 4-\log (1+\sqrt{3} \tan x)) d x \\ \Rightarrow I =\log 4 \cdot \frac{\pi}{3}- I \\ \Rightarrow I =\frac{\pi}{6} \log 4=\frac{\pi}{3} \log 2 . \end{array} $