A) \[\frac{3\pi }{64}\]
B) \[\frac{3\pi }{572}\]
C) \[\frac{3\pi }{256}\]
D) \[\frac{3\pi }{128}\]
Correct Answer: C
Solution :
\[I=\int_{-\pi /2}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx}\]\[=2\int_{0}^{^{\pi /2}}{{{\sin }^{4}}x\,{{\cos }^{6}}x.\,dx}\] \[\begin{matrix} \because \int_{-a}^{a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}} & \text{if }f(-x)=f(x) \\ \,\,\,\,\,=0, & \text{if }f(-x)=-f(x) \\ \end{matrix}\] Applying Gamma function, we get \[I=\frac{2\,\Gamma 5/2\,.\,\Gamma 7/2}{2\,.\Gamma 6}\] \[=\frac{3/2.1/2.\sqrt{\pi .}5/2.3/2.1/2.\sqrt{\pi }}{5.4.3.2.1}\]\[=\frac{3\pi }{{{2}^{8}}}=\frac{3\pi }{256}\].
You need to login to perform this action.
You will be redirected in
3 sec
