Definite Integration Question 108
Question: $ \int _{\pi /3}^{\pi /2}{\frac{\sqrt{1+\cos x}}{{{(1-\cos x)}^{\frac{5}{2}}}}}dx= $
[AI CBSE 1980]
Options:
A) $ \frac{5}{2} $
B) $ \frac{3}{2} $
C) $ \frac{1}{2} $
D) $ \frac{2}{5} $
Show Answer
Answer:
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}}=| \frac{{t^{-2}}}{-2} |} _{1/2}^{1}=\frac{3}{2} $ .
BETA
