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

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(B) माना $I = \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(i)$गुणधर्म $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$ का उपयोग करने पर:$I =

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(B) माना $I = \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(i)$गुणधर्म $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$ का उपयोग करने पर:$I = \int_{0}^{\pi} e^{\sin^2(\pi - x)} \cos^3(\pi - x) \, dx$चूंकि $\sin(\pi - x) = \sin x$ और $\cos(\pi - x) = -\cos x$,इसलिए:$I = \int_{0}^{\pi} e^{\sin^2 x} (-\cos x)^3 \, dx$$I = - \int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ ... $(ii)$$(i)$ और $(ii)$ को जोड़ने पर:$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$

$\int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ का मान क्या है?

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