A) \[\frac{20}{21}\]
B) \[\frac{8}{21}\]
C) \[\frac{-20}{21}\]
D) \[\frac{-8}{21}\]
Correct Answer: B
Solution :
Let \[I=\int_{0}^{\pi /2}{\sqrt{\cos \theta }}{{\sin }^{3}}\theta \,\,d\theta \] Put \[t=\cos \theta \Rightarrow dt=-\sin \theta \,\,d\theta ,\] then I =\[-\int_{1}^{0}{{{t}^{1/2}}(1-{{t}^{2}})dt=\int_{0}^{1}{({{t}^{1/2}}-{{t}^{5/2}})}}\]\[dt\] I = \[\left[ \frac{2}{3}{{t}^{3/2}}-\frac{2}{7}{{t}^{7/2}} \right]_{0}^{1}=\frac{8}{21}\].
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