Solving an integral with exponential function

Question

I try to solve the following integral $$\int_a^b \exp\left\{-\lambda \left(\frac{y}{2x^2}-\frac{1}{x}\right)\right\} dx$$ for $\lambda>0$ and $y \in \mathbb{R}$. Do you see any relation to any known integral or another way to solve it?

Thanks a lot!!

Answer 1

let $t=\frac1x-\frac2y$, we get $$e^{\frac{2\lambda}y}\int_{\frac1b-\frac2y}^{\frac1a-\frac2y}e^{-\frac{\lambda y}2t^2}\left(t+\frac2y\right)^2dt,$$ because there exist an integral like $$\int e^{-x^2}x^2dx=-\frac12xe^{-x^2}+\frac12\int e^{-x^2}dx,$$ so ...