Question

$\underset{-\log 3}{\overset{\log 3}{\int }}{\cot }^{-1}\left(\frac{e^x-1}{e^x+1}\right)dx=$ _____

• A. $\frac{\pi }{2}\log 3$
• B. $\pi \log 3$
• C. $0$
• D. $\pi \log 9$

Answer 1

The correct option is B $\pi \log 3$

$I=\underset{-\log 3}{\overset{\log 3}{\int }}{\cot }^{-1}\left(\frac{e^x-1}{e^x+1}\right)dx$

$\Rightarrow I=\underset{-\log 3}{\overset{\log 3}{\int }}{\cot }^{-1}\left(\frac{e^{-x}-1}{e^{-x}+1}\right)dx$

$\Rightarrow I={\int }_{-\log 3}^{\log 3}{\cot }^{-1}\left(\frac{1-e^x}{1+e^x}\right)dx$

$\Rightarrow I={\int }_{-\log 3}^{\log 3}(\pi -{\cot }^{-1}\left(\frac{e^x-1}{1+e^x}\right))dx$

$\Rightarrow 2I=\pi \underset{-\log 3}{\overset{\log 3}{\int }}dx$
$\Rightarrow I=\frac{1}{2}\pi \left(2\log 3\right)$
$\Rightarrow I=\pi \log 3$