A) \[5\pi /8\]
B) \[5\pi /16\]
C) \[5\pi /2\]
D) \[5\pi /4\]
E) \[5\pi /32\]
Correct Answer: D
Solution :
Given that, \[\int_{0}^{\pi /2}{{{\sin }^{6}}x}dx=\frac{5\pi }{32}\] Let \[I=\int_{-\pi }^{\pi }{({{\sin }^{6}}x+{{\cos }^{6}}x})dx\] \[=2\int_{0}^{\pi }{({{\sin }^{6}}x+{{\cos }^{6}}x})dx\] \[=4\int_{0}^{\pi /2}{({{\sin }^{6}}x+{{\cos }^{6}}x})dx\] \[=4\int_{0}^{\pi /2}{{{\sin }^{6}}x\,dx+4\int_{0}^{\pi /2}{{{\cos }^{6}}x}}\left( \frac{\pi }{2}-x \right)dx\] \[=8\int_{0}^{\pi /2}{{{\sin }^{6}}x\,dx+8}\times \frac{5.3.1}{6.4.2}\times \frac{\pi }{2}\] \[=\frac{5\pi }{4}\]
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