Integrating exponential function raised to a fractional term

Question

I'm trying to integrate an exponential term raised to a fractional power with other variables in it. I'm really rusty and having a hard time trying to figure out where to start. I'd like to pull out the other terms so I can work on integrating but I'm not sure if or how I can do that with it being part of the exponent. Below is the what I am trying to integrate.

$$\int_e{e^{\frac{-x^2\beta^2-y^22\alpha^2}{2\alpha^2\beta^2}}dy}$$

Any help is greatly appreciated.

Answer 1

Indefinite integral would be: $$\int{e^{\frac{-x^2\beta^2-y^22\alpha^2}{2\alpha^2\beta^2}}dy}$$=$$\int{e^{\frac{-y^22\alpha^2}{2\alpha^2\beta^2}}e^{\frac{-x^2\beta^2}{2\alpha^2\beta^2}}dy}$$=$$\int{e^{\frac{-y^2}{\beta^2}}e^{\frac{-x^2}{2\alpha^2}}dy}$$=$$e^{\frac{-x^2}{2\alpha^2}}\int{e^{\frac{-y^2}{\beta^2}}dy}$$=$$e^{\frac{-x^2}{2\alpha^2}}(\frac{\sqrt\pi\beta\ erf(\frac{y}{\beta})}{2} +C)$$ So just put the limits and you will get the answer you were looking for.

Answer 2

we ge this here $$1/2\,{{\rm e}^{-1/2\,{\frac {{x}^{2}}{{\beta}^{2}}}}}\sqrt {\pi } \sqrt {2}\alpha\, {{\rm erf}\left(1/2\,{\frac {\sqrt {2}y}{\alpha}}\right)} $$