int_0^{pi / 2} e^{sin x} cdot cos x , dx = | vedclass

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(C) Let $I = \int_0^{\pi / 2} e^{\sin x} \cos x \, dx$.Substitute $\sin x = t$. Then,differentiating both sides with respect to $x$,we get $\cos x \,

Vedclass · Accepted Answer

$e-1$

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(C) Let $I = \int_0^{\pi / 2} e^{\sin x} \cos x \, dx$.Substitute $\sin x = t$. Then,differentiating both sides with respect to $x$,we get $\cos x \, dx = dt$.Now,change the limits of integration:When $x = 0$,$t = \sin(0) = 0$.When $x = \frac{\pi}{2}$,$t = \sin(\frac{\pi}{2}) = 1$.Substituting these into the integral,we get:$I = \int_0^1 e^t \, dt$.The integral of $e^t$ is $e^t$.$I = [e^t]_0^1 = e^1 - e^0 = e - 1$.

$\int_0^{\pi / 2} e^{\sin x} \cdot \cos x \, dx =$

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