calculus - Improper integral $ int_{0}^{infty } e^{-sqrt{x}}text dx$

Help me please with this improper integral:

$ \int_{0}^{\infty } e^{-\sqrt{x}}\text dx$

Thanks.

I solved it partially, and stuck after integration by parts.

Answer

$$I= \displaystyle \lim_{a \to \infty} \int \limits_{0}^{a} e^{-\sqrt{x}}\, \text dx= \displaystyle \lim_{a \to \infty} \left(2-2 \cdot\frac{\sqrt{a}+1}{e^{\sqrt a}}\right)=2-2\cdot \displaystyle \lim_{a \to \infty} \frac{\sqrt{a}+1}{e^{\sqrt a}}=2$$

The last limit can be evaluated using substitution $t=\sqrt{a}~$ and L'Hopital rule .