Evaluate the following definite integrals: _ 0 ^ ^ -x dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{\infty}\text{e}^{-\text{x}}\text{ dx}$

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Answer

We have$\int_{0}^\limits{\infty}\text{e}^{-\text{x}}\text{ dx}$
We know that $\int\text{e}^{-\text{x}}=-\text{e}^{-\text{x}}$$\int_{0}^\limits{\infty}\text{e}^{-\text{x}}\text{ dx}$
$=\big[-\text{e}^{-\text{x}}\big]^{\infty}_0$
$=\big[\text{e}^{-\infty}+\text{e}^{-0}\big]$ $[\because\text{e}^{\infty}=0,\text{ e}^0=1\big]$
$=[-0+1]$
$=1$

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