Help in evaluating an integral of exponential function

Question

I am trying to evaluate the following integral $$ I = \int_{0}^{t}s^{-b-1}e^{-\frac{1}{2} a^2 s^{-2 b}} ds$$ where $a > 0$ and $ 0 \le b \le 1$.

I am not quite sure how to solve this. Any help would be much appreciated.

Answer 1

HINT:

Enforce the substitution $s\to s^{-1/b}$ to yield the integral

$$\frac1b \int_{t^{-1/b}}^\infty e^{-\frac12 a^2s^s}\,ds$$