A) \[\frac{5}{2}\]
B) \[\frac{3}{2}\]
C) \[\frac{1}{2}\]
D) \[\frac{2}{5}\]
Correct Answer: B
Solution :
\[I=\int_{\pi /3}^{\pi /2}{\frac{\sqrt{1+\cos x}}{{{(1-\cos x)}^{5/2}}}\times \frac{\sqrt{1-\cos x}}{\sqrt{1-\cos x}}}\,\,dx\] = \[\int_{\pi /3}^{\pi /2}{\,\,\frac{\sin x}{{{(1-\cos x)}^{3}}}\,dx}\] Now, put \[1-\cos x=t\] Also, when \[x=\frac{\pi }{3},t=\frac{1}{2}\] and \[x=\frac{\pi }{2}\,,\,\,t=1\] Therefore, \[I=\int_{1/2}^{1}{\frac{dt}{{{t}^{3}}}=\left| \frac{{{t}^{-2}}}{-2} \right|}_{1/2}^{1}=\frac{3}{2}\].
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