Definite Integration Question 270
Question: $ \int _{-\pi /2}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}xdx=} $
[EAMCET 2002]
Options:
A) $ \frac{3\pi }{64} $
B) $ \frac{3\pi }{572} $
C) $ \frac{3\pi }{256} $
D) $ \frac{3\pi }{128} $
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Answer:
Correct Answer: C
Solution:
$ I=\int _{-\pi /2}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}xdx} $
$ =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} $ .
BETA
