A) \[\frac{3}{8}\]
B) \[\frac{1}{8}\]
C) \[-\frac{3}{8}\]
D) None of these
Correct Answer: A
Solution :
Putting \[x=\tan \theta \], we get \[\int_{0}^{\infty }{\frac{dx}{{{\left( x+\sqrt{{{x}^{2}}+1} \right)}^{3}}}}\] \[=\int_{0}^{\pi /2}{\frac{{{\sec }^{2}}\theta \,d\theta }{{{(\tan \theta +\sec \theta )}^{3}}}}=\int_{0}^{\pi /2}{\frac{\cos \theta }{{{(1+\sin \theta )}^{3}}}d\theta }\] \[=\left[ -\frac{1}{2{{(1+\sin \theta )}^{2}}} \right]_{0}^{\pi /2}=-\frac{1}{8}+\frac{1}{2}=\frac{3}{8}\].
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