Integration that look like incomplete Gamma function

Question

I am trying to solve the following integration $$\int_0^cx^{-k}e^{-gx^2}dx$$ where $c,k$ and $g$ are positive values. I know how to solve it if there were $k$ instead of $-k$. I need help with this integration.

Thanks in advance

Answer 1

Considering $$I=\int x^{-k}e^{-gx^2}dx$$ change variable $$g x^2=t\implies x=\frac{\sqrt{t}}{\sqrt{g}}\implies dx=\frac{1}{2 \sqrt{g} \sqrt{t}}$$ All of that makes $$I=\frac{1}{2}g^{\frac{k-1}{2}}\int t^{-\frac{k+1}{2}}e^{-t}\,dt=-\frac{1}{2}g^{\frac{k-1}{2}}\,\Gamma \left(\frac{1-k}{2},t\right)$$ But for the definite integral, a requirement is $\Re(k)<1$.