Integral of this exponential function: does it have a solution?

[fchopin]

Hi all,

I'm trying to solve the definite integral between 0 and inf of:

exp(a*x^2 + b*x + c)
--------------------- dx
1 + exp(m*x + n)

with a,b,c,m,n real numbers and a < 0 (negative number so it converges).

I've read in the forum's rules that I have to post the work that I have done to get an answer but I have nothing reasonable to post (I have tried many alternatives but I didn't suceed, sorry)

A way to obtain the exact solution would be perfect but an approximate result, even an upper/lowerbound would be fine as well.

Any idea or help, please?

Thanks in advance,
FC.

 

[g_edgar]

Can you do the special case...
$$\int _{0}^{\infty }\!{\frac {{{\rm e}^{-{x}^{2}}}}{1+{{\rm e}^{x}}}}{d<br /> x}$$

 

[Pere Callahan]

Can you do the special case...
$$\int _{0}^{\infty }\!{\frac {{{\rm e}^{-{x}^{2}}}}{1+{{\rm e}^{x}}}}{d<br /> x}$$

Mathematica can't...

 

[Gib Z]

Maybe he's alluding to the idea that if you can't do one of the basic cases you wouldn't be able to do the general case either.

 

[fchopin]

Hi guys,

thank you for your answers. I have found that if m<0 and n<0 then 1/(1 + exp(m*x + n)) can be expanded into Maclaurin series, which yields something like 1 + e^() - e^2*(). The terms can be multiplied by the numerator of the original integral, thus finally obtaining three integrals of the form exp^(a*x^2 + b*x + c), which have closed-form solutions in terms of the error function.

Thank you for your interest!