int_0^pi x sin^7 x cos^6 x , dx = | vedclass

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(C) Let $I = \int_0^\pi x \sin^7 x \cos^6 x \, dx$. Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we have: $I = \int_0^\pi (\pi - x

Vedclass · Accepted Answer

$\frac{16 \pi}{3003}$

Vedclass · Answer

(C) Let $I = \int_0^\pi x \sin^7 x \cos^6 x \, dx$. Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we have: $I = \int_0^\pi (\pi - x) \sin^7(\pi - x) \cos^6(\pi - x) \, dx$ Since $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$,we get: $I = \int_0^\pi (\pi - x) \sin^7 x (-\cos x)^6 \, dx = \int_0^\pi (\pi - x) \sin^7 x \cos^6 x \, dx$ $I = \pi \int_0^\pi \sin^7 x \cos^6 x \, dx - I$ $2I = \pi \int_0^\pi \sin^7 x \cos^6 x \, dx$ Using $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ if $f(2a-x) = f(x)$,we have: $I = \pi \int_0^{\pi/2} \sin^7 x \cos^6 x \, dx$ Using Wallis' Formula $\int_0^{\pi/2} \sin^m x \cos^n x \, dx = \frac{(m-1)!!(n-1)!!}{(m+n)!!}$: $I = \pi 	imes \frac{6 	imes 4 	imes 2 	imes 5 	imes 3 	imes 1}{13 	imes 11 	imes 9 	imes 7 	imes 5 	imes 3 	imes 1} = \pi 	imes \frac{48}{13 	imes 11 	imes 9 	imes 7} = \pi 	imes \frac{16}{13 	imes 11 	imes 3 	imes 7} = \frac{16 \pi}{3003}$.

$\int_0^\pi x \sin^7 x \cos^6 x \, dx =$

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