Definite Integration Question 263
Question: The value of $ \int _{0}^{\pi /2}{{{( \sqrt{\sin \theta }\cos \theta )}^{3}}d\theta } $ is
[AMU 1999]
Options:
A) 2/9
B) 2/15
C) 8/45
D) 5/2
Show Answer
Answer:
Correct Answer: C
Solution:
$ \int _{0}^{\pi /2}{{{(\sqrt{\sin \theta }\cos \theta )}^{3}}d\theta =\int _{0}^{\pi /2}{{{\sin }^{3/2}}\theta {{\cos }^{3}}\theta d\theta }} $
Applying gamma function, $ \int_0^{\pi /2}{{{\sin }^{3/2}}\theta {{\cos }^{3}}\theta d\theta } $
$ =\frac{\Gamma ( \frac{\frac{3}{2}+1}{2} )\Gamma ( \frac{3+1}{2} )}{2\Gamma ( \frac{\frac{3}{2}+3+2}{2} )} $
$ =\frac{\Gamma (5/4)\Gamma 2}{2\Gamma (13/4)} $
$ =\frac{\Gamma ( \frac{5}{4} )}{2.\frac{9}{4}.\frac{5}{4}.\Gamma ( \frac{5}{4} )} $
$ =\frac{8}{45} $ .
BETA
