Help for solving $ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx $

Question

Salam,

I would appreciate it if anyone could help me solving this integral:

$$ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx $$

Many thanks.

Answer 1

The indefinite integral cannot be expressed in terms of elementary functions. However, the definite integral can, for appropriate limits, and $b=0$, be expressed in terms of special Bessel I functions:

$$\int_{-1}^1\dfrac{e^{-ax^2}}{\sqrt{1-x^2}}dx~=~2\int_0^1\dfrac{e^{-ax^2}}{\sqrt{1-x^2}}dx~=~\pi~\exp\bigg(\dfrac a2\bigg)~I_0\bigg(\dfrac a2\bigg).$$

More information can be found here.

Answer 2

The solution to this integral cannot be found in terms of elementary mathematical functions.