The value of int_0^pi sin^{50} x cos^{49} x , | vedclass

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(A) Let $I = \int_0^\pi \sin^{50} x \cos^{49} x \, dx$. Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we have: $I = \int_0^\pi \sin

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(A) Let $I = \int_0^\pi \sin^{50} x \cos^{49} x \, dx$. Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we have: $I = \int_0^\pi \sin^{50}(\pi - x) \cos^{49}(\pi - x) \, dx$. Since $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$,we get: $I = \int_0^\pi \sin^{50} x (-\cos x)^{49} \, dx$. $I = -\int_0^\pi \sin^{50} x \cos^{49} x \, dx$. $I = -I$. Adding $I$ to both sides,we get $2I = 0$,which implies $I = 0$.

The value of $\int_0^\pi \sin^{50} x \cos^{49} x \, dx$ is

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