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  4. If ∫0π /2 sin 6xdx=(5π /32), then the value of ∫-π π ( sin 6x+ cos 6x)dx is

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Q. If $ \int_{0}^{\pi /2}{{{\sin }^{6}}x}dx=\frac{5\pi }{32}, $ then the value of $ \int_{-\pi }^{\pi }{({{\sin }^{6}}x+{{\cos }^{6}}x)}dx $ is

KEAMKEAM 2007Integrals

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} $