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[Yiping]

I try to find the solution of the integrals below for more than a week.
\int_{p>0}^{\infty}\frac{\exp[-ap]}{p}dp
can anyone give me a hint about how to solve it?

[CompuChip]

e^{-a p} is continuous on ]0, 1] and therefore bounded (it lies between m and M). So you can estimate it to this constant and then on ]0, 1] the integral diverges logarithmically (\displaystyle\propto \int_0^1 dp/p). Furthermore, the integrand is positive on ]1, infinity[ so the end result also diverges.

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That is definitely true...