A) \[{{\log }_{e}}\left( \frac{2}{3} \right)\]
B) \[{{\log }_{e}}3\]
C) \[\frac{1}{2}{{\log }_{e}}\left( \frac{4}{3} \right)\]
D) \[{{\log }_{e}}\left( \frac{4}{3} \right)\]
Correct Answer: D
Solution :
Put \[1+\tan x=t\Rightarrow {{\sec }^{2}}x\,dx=dt\] \[\therefore \,\,\,\int_{0}^{\pi /4}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}dx}\] \[=\int_{1}^{2}{\frac{dt}{t(1+t)}}=\int_{1}^{2}{\frac{dt}{t}-\int_{1}^{2}{\frac{dt}{1+t}}}=[\log t-\log (1+t)]_{1}^{2}\] \[={{\log }_{e}}2-{{\log }_{e}}3+{{\log }_{e}}2={{\log }_{e}}\frac{4}{3}\].
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