int_{0}^{frac{pi}{2}}left(e^{sin x}-e^{cos x} | vedclass

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} (e^{\sin x} - e^{\cos x}) dx$ ....$(1)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} (e^{\sin x} - e^{\cos x}) dx$ ....$(1)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get:$I = \int_{0}^{\frac{\pi}{2}} (e^{\sin(\frac{\pi}{2}-x)} - e^{\cos(\frac{\pi}{2}-x)}) dx$$I = \int_{0}^{\frac{\pi}{2}} (e^{\cos x} - e^{\sin x}) dx$ ....$(2)$Adding equation $(1)$ and $(2)$:$2I = \int_{0}^{\frac{\pi}{2}} (e^{\sin x} - e^{\cos x} + e^{\cos x} - e^{\sin x}) dx$$2I = \int_{0}^{\frac{\pi}{2}} (0) dx$$2I = 0 \Rightarrow I = 0$

$\int_{0}^{\frac{\pi}{2}}\left(e^{\sin x}-e^{\cos x}\right) d x=$

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