int_0^{2 pi} sin ^6 x cos ^5 x , dx is equal | vedclass

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(C) Let $I = \int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$. Using the property $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ if $f(2a-x) = f(x)$, we ch

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(C) Let $I = \int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$. Using the property $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ if $f(2a-x) = f(x)$, we check $f(x) = \sin^6 x \cos^5 x$. Since $f(2\pi - x) = \sin^6(2\pi - x) \cos^5(2\pi - x) = (-\sin x)^6 (\cos x)^5 = \sin^6 x \cos^5 x = f(x)$, we have $I = 2 \int_0^{\pi} \sin^6 x \cos^5 x \, dx$. Now, using the property $\int_0^a f(x) \, dx = 0$ if $f(a-x) = -f(x)$, we check $f(\pi - x)$. $f(\pi - x) = \sin^6(\pi - x) \cos^5(\pi - x) = (\sin x)^6 (-\cos x)^5 = -\sin^6 x \cos^5 x = -f(x)$. Therefore, $\int_0^{\pi} \sin^6 x \cos^5 x \, dx = 0$. Hence, $I = 2 	imes 0 = 0$.

$\int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$ is equal to

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