int_{0}^{pi} e^{sin^2 x} cos^3 x , dx is equa | vedclass

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(B) Let $I = \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(i)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I =

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(B) Let $I = \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(i)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0}^{\pi} e^{\sin^2(\pi - x)} \cos^3(\pi - x) \, dx$Since $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$,we have:$I = \int_{0}^{\pi} e^{\sin^2 x} (-\cos x)^3 \, dx$$I = - \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(ii)$Adding $(i)$ and $(ii)$,we get:$I + I = \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx - \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$$2I = 0$$I = 0$

$\int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ is equal to

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