# How to integrate $\int_0^t \frac{1}{x^{w/2}} e^{-c/x} dx$?

I've come up with the following integral as part of an equation for a generalized flow model for groundwater flow. Unfortunately it doesn't converge. Does anyone know of a way I can manipulate it in order to make it solvable?

$$\int_0^t \frac{1}{x^{w/2}} e^{-c/x} dx$$

$$1 \leq w \le 3$$ and $$c$$ is a positive number

We make the transformation: $$y=\frac{c}{ x}$$. We have: $$dx=-\frac{cdy}{ y^2}$$, and we can put the integral in the form: $$I=c^{1-\frac{w}{2}}\int_{\frac{c}{t}}^\infty y^{\frac{w}{2}-2}e^{-y}dy=c^{1-\frac{w}{2}}\Gamma\bigg(\frac{w}{2}-1,\frac{c}{t}\bigg)$$ where $$\Gamma$$ is the incomplete gamma function.
• I think I see now – by definition $$\int_{x}^\infty y^{a-1}e^{-y}dy= \Gamma\bigg(a,x\bigg)$$ The $c^{3-\frac{w}{2}}$ I’m not sure about though. – N A Oct 30 '20 at 21:45