Definite Integration Question 249
Question: $ \int_0^{\infty }{\frac{dx}{{{( x+\sqrt{x^{2}+1} )}^{3}}}}= $
[EAMCET 1992]
Options:
A) $ \frac{3}{8} $
B) $ \frac{1}{8} $
C) $ -\frac{3}{8} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Putting $ x=\tan \theta $ , we get $ \int_0^{\infty }{\frac{dx}{{{( x+\sqrt{x^{2}+1} )}^{3}}}} $
$ =\int_0^{\pi /2}{\frac{{{\sec }^{2}}\theta d\theta }{{{(\tan \theta +\sec \theta )}^{3}}}}=\int_0^{\pi /2}{\frac{\cos \theta }{{{(1+\sin \theta )}^{3}}}d\theta } $
$ =[ -\frac{1}{2{{(1+\sin \theta )}^{2}}} ]_0^{\pi /2}=-\frac{1}{8}+\frac{1}{2}=\frac{3}{8} $ .
BETA
