A) 2/9
B) 2/15
C) 8/45
D) 5/2
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, \[\int_{0}^{\pi /2}{{{\sin }^{3/2}}\theta {{\cos }^{3}}\theta d\theta }\]\[=\frac{\Gamma \left( \frac{\frac{3}{2}+1}{2} \right)\Gamma \left( \frac{3+1}{2} \right)}{2\Gamma \left( \frac{\frac{3}{2}+3+2}{2} \right)}\] \[=\frac{\Gamma (5/4)\,\,\Gamma \,2}{2\Gamma \,(13/4)}\]\[=\frac{\Gamma \,\left( \frac{5}{4} \right)}{2.\frac{9}{4}.\frac{5}{4}.\Gamma \left( \frac{5}{4} \right)}\]\[=\frac{8}{45}\].
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