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

Question

Dr. Abubacker · Accepted Answer

(d)
Let $I = \int_{- \pi}^{\pi}{(\sin^{6}x} + \cos^{6}x)\ dx = 2\int_{0}^{\pi}{(\sin^{6}x} + \cos^{6}x)\ dx$
$= 4\int_{0}^{\frac{\pi}{2}}{\sin^{6}x}\ dx + 4\int_{0}^{\frac{\pi}{2}}{\cos^{6}\left( \frac{\pi}{2} - x ight)}\ dx$
$= 8\int_{0}^{\frac{\pi}{2}}{\sin^{6}x}\ dx = 8 	imes \frac{5.3.1}{6.4.2} 	imes \frac{\pi}{2} = \frac{5\pi}{4}$

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

Explanation

Reviewed by Dr. Abubacker

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