Definite Integration Question 460
Question: $ \int_0^{\pi /4}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}}dx= $
Options:
A) $ {\log _{e}}( \frac{2}{3} ) $
B) $ {\log _{e}}3 $
C) $ \frac{1}{2}{\log _{e}}( \frac{4}{3} ) $
D) $ {\log _{e}}( \frac{4}{3} ) $
Show Answer
Answer:
Correct Answer: D
Solution:
Put $ 1+\tan x=t\Rightarrow {{\sec }^{2}}xdx=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} $ .
BETA
