Answer
The integral converges to: $\dfrac{-1}{a}; a \lt 0$
Work Step by Step
We will compute the integral as follows:
$\int_{0}^{\infty} e^{ax} \ dx= \lim\limits_{R \to \infty} \int_{0}^R e^{ax} \ dx \\=\lim\limits_{R \to \infty} \dfrac{e^{ax}}{a}|_0^R \\=\lim\limits_{R \to \infty} [\dfrac{e^{aR}}{a}-\dfrac{1}{a}]\\=0-\dfrac{1}{a}\\=-\dfrac{1}{a}$
Hence, the integral converges to:
$\int_{0}^{\infty} e^{ax} \ dx =\dfrac{-1}{a}; a \lt 0$
