int_{0}^{pi} cos^3 x , dx = | vedclass

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(B) Let $I = \int_{0}^{\pi} \cos^3 x \, dx$. Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we have: $I = \int_{0}^{\pi} \co

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$0$

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(B) Let $I = \int_{0}^{\pi} \cos^3 x \, dx$. Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we have: $I = \int_{0}^{\pi} \cos^3(\pi - x) \, dx$. Since $\cos(\pi - x) = -\cos x$,then $\cos^3(\pi - x) = -\cos^3 x$. So,$I = \int_{0}^{\pi} -\cos^3 x \, dx = -I$. $2I = 0 \implies I = 0$. Alternatively,using the property $\int_{0}^{2a} f(x) \, dx = 0$ if $f(2a-x) = -f(x)$,here $f(x) = \cos^3 x$ and $2a = \pi$,so $f(\pi - x) = -f(x)$,thus the integral is $0$.

$\int_{0}^{\pi} \cos^3 x \, dx = $

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