Definite-Integration Question 504
Question: $ \int_0^{\infty }{\frac{\log ,(1+x^{2})}{1+x^{2}}},dx= $
Options:
A) $ \pi \log \frac{1}{2} $
B) $ \pi \log 2 $
C) $ 2\pi \log \frac{1}{2} $
D) $ 2\pi \log 2 $
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ I=\int_0^{\infty }{\frac{\log (1+x^{2})}{1+x^{2}}dx} $ Put $ x=\tan \theta \Rightarrow dx={{\sec }^{2}}\theta ,d\theta , $
$ \therefore $ $ I=\int_0^{\pi /2}{\log {{(\sec \theta )}^{2}}d\theta =2\int_0^{\pi /2}{\log \sec \theta d\theta }} $ $ =-2\int_0^{\pi /2}{\log \cos \theta d\theta =-2.\frac{\pi }{2}\log \frac{1}{2}} $ $ =-\pi \log \frac{1}{2}=\pi \log 2 $ .
BETA
