\(\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}}\left[\frac{\tan x}{\tan x+\cot x}\right] d x=\)

(C)
\(\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}}\left[\frac{\tan x}{\tan x+\cot x}\right] d x...(1) \)
\( =\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}}\left[\frac{\tan \left(\frac{3 \pi}{10}+\frac{\pi}{5}-x\right)+\cot \left(\frac{3 \pi}{10}+\frac{\pi}{5}-x\right)}{\tan \left(\frac{3 \pi}{10}+\frac{\pi}{5}-x\right)}\right] d x=\) \(\int_{\frac{\pi}{5}}^{10} \frac{3 \pi}{\tan \left(\frac{\pi}{2}-x\right)+\cot \left(\frac{\pi}{2}-x\right)} \tan \left(\frac{\pi}{2}-x\right) \)
\( =\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}} \frac{\tan x+\cot x} d x...(2)\)
Equation (1) + (2) gives
\(2 I =\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}} d x=[x]_{\frac{\pi}{5}}^{\frac{3 \pi}{10}}=\left(\frac{3 \pi}{10}-\frac{\pi}{5}\right)=\frac{\pi}{10} \)
\( \therefore I =\frac{\pi}{20}\)