# How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that

$$\int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1}{p}\right)\,\,\mathrm{d}x = p(1-e^{-\epsilon})$$

I am interested in calculating the related integral

$$\int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1-\delta x}{p-\delta x}\right)\,\mathrm{d}x.$$

I have tried Mathematica to no avail. I would appreciate any help. I would love to hear the reasoning behind the steps if someone has a solution.

Thanks a lot!

• Are you asking for a closed form or an asymptotic? – Igor Rivin Oct 13 '14 at 20:28
• A closed form would be great, if one exists. If not, understanding asymptotic behavior would be very helpful. – Mert Sağlam Oct 13 '14 at 20:30
• Have you tried steepest descent for asymptotics? Also are you interested in asymptotics for large $p$ or $\delta$ – Alex R. Oct 13 '14 at 20:35
• Thanks, I am looking up steepest descent, I have not used it before. I am interested in the behavior as $\epsilon$ gets smaller. – Mert Sağlam Oct 13 '14 at 20:53
• Mathematica gives the series expansion $p\epsilon+O(\epsilon^2)$. – Stopple Oct 13 '14 at 21:07