Definite Integration Question 114
Question: $ \int_0^{\pi /2}{{{( \frac{\theta }{\sin \theta } )}^{2}}d\theta =} $
Options:
A) $ \pi \log 2 $
B) $ \frac{\pi }{\log 2} $
C) $ \pi $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ I=\int_0^{\pi /2}{{{( \frac{\theta }{\sin \theta } )}^{2}}d\theta }=[-{{\theta }^{2}}\cot \theta ]_0^{\pi /2}+\int_0^{\pi /2}{2\theta .\cot \theta .d\theta } $
$ =2[\theta .\log \sin \theta ]_0^{\pi /2}-2\int_0^{\pi /2}{\log \sin \theta d\theta } $
$ \Rightarrow \frac{I}{2}=0-\underset{\theta \to 0}{\mathop{\lim }}\theta \log .\sin \theta $
$ -\int_0^{\pi /2}{\log \sin \theta d\theta } $
therefore $ \frac{\pi }{2}\log 2 $ .
Hence I = $ \pi \log 2 $ .
BETA
