# Integral of this exponential function: analytical solution?

• October 31st 2009, 02:50 PM
fryderykchopin
Integral of this exponential function: analytical solution?
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, m < 0 (negative number so it converges).

I have tried to transform the denominator to cosh and integrate by parts,
among many others alternatives but I didn't suceed.

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

FC.
• November 1st 2009, 12:00 AM
NonCommAlg
Quote:

Originally Posted by fryderykchopin
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, m < 0 (negative number so it converges).

I have tried to transform the denominator to cosh and integrate by parts,
among many others alternatives but I didn't suceed.

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

if we also assume that $n \leq 0,$ then $\frac{1}{1+e^{mx+n}} \approx 1 - e^{mx + n} + e^{2mx+2n}$ and thus: $\text{your integral}$ $\approx \int_0^{\infty}e^{ax^2+bx+c} \ dx \ - \ \int_0^{\infty} e^{ax^2+(b+m)x + c+n} \ dx \ + \ \int_0^{\infty} e^{ax^2+(b+2m)x + c+2n} \ dx.$