Evaluate: ^ /2 _0e^x( x- x)dx. - Vidyadip

Question

Evaluate: $\int\limits^{\pi/2}_0\text{e}^x(\sin x-\cos x)\text{d}x.$

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Answer

Let I =$\int\limits^{\pi/2}_0\text{e}^x(\sin x-\cos x)\text{ d}x$
Using f(a) = f(a-x), we have:
$\therefore\ \ \ \int\limits^{\pi/2}_0\text{e}^x(\cos x-\sin x)\text{ d}x\int\limits^{\pi/2}_0\text{e}^x(\cos x+(-\sin x))\text{ d}x$
we know that, $\int\text{e}^x[f(x)+f'(x)]dx=\text{e}^xf(x)+\text{C}$
$\text{Also},\frac{d\cos x}{dx}=-\sin x$
$\text{So, I}=\text{e}^x\cos x^{\pi/2}_0$
$\Rightarrow\text{I}=(\text{e}^{\pi/2}\times0)-(\text{e}^0\times1)=0-1=-1$
$\therefore\ \ \int\limits^{\pi/2}_0\text{e}^x(\sin x-\cos x)\text{ d}x=-1$

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