Evaluate the following integrals: ^ 2 _0e^ x ( - )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\text{e}^{\text{x}}(\sin\text{x}-\cos\text{x})\text{dx}$

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Answer

$\int\limits^{\frac{\pi}{2}}_0\text{e}^{\text{x}}(\sin\text{x}-\cos\text{x})\text{dx}$
$=-\int\limits^{\frac{\pi}{2}}_0\text{e}^{\text{x}}\Big[\cos\text{x}+(-\sin\text{x})\Big]\text{dx}$
$=-\big[\text{e}^{\text{x}}\cos\text{x}\big]^{\frac{\pi}{2}}_0$ $\Big\{\int\text{e}^{\text{x}}\big[\text{f(x)}+\text{f}'(\text{x})\big]\text{dx}=\text{e}^{\text{x}}\text{f(x)}+\text{C}\Big\}$
$=-\Big(\text{e}^{\frac{\pi}{2}}\cos\frac{\pi}{2}-\text{e}^0\cos0\Big)$
$=-\Big(\text{e}^{\frac{\pi}{2}}\times0-1\times1\Big)$
$=-(0-1)$
$=1$

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