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

In summary, the conversation discusses the problem of solving the definite integral of a given function between 0 and infinity. The function has a special case where the integral can be expanded into Maclaurin series, leading to a closed-form solution using the error function. The participants are looking for help or ideas to solve the integral.
  • #1
fchopin
10
0
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.
 
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  • #2
Can you do the special case...
[tex]
\int _{0}^{\infty }\!{\frac {{{\rm e}^{-{x}^{2}}}}{1+{{\rm e}^{x}}}}{d
x}
[/tex]
 
  • #3
g_edgar said:
Can you do the special case...
[tex]
\int _{0}^{\infty }\!{\frac {{{\rm e}^{-{x}^{2}}}}{1+{{\rm e}^{x}}}}{d
x}
[/tex]

Mathematica can't...
 
  • #4
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.
 
  • #5
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!
 

1. What is the integral of an exponential function?

The integral of an exponential function is the inverse operation of differentiation. It is a mathematical concept used to find the area under the curve of an exponential function.

2. Is there a general formula for the integral of an exponential function?

Yes, the general formula for the integral of an exponential function is f(x) = ae^(bx) is F(x) = (a/b)e^(bx) + C, where C is the constant of integration.

3. Can an exponential function always be integrated?

Yes, exponential functions can always be integrated, as long as the function is continuous and defined on the interval of integration.

4. How do you evaluate the definite integral of an exponential function?

To evaluate the definite integral of an exponential function, you can use the general formula and substitute the upper and lower limits of integration into the equation. Then, subtract the value of the function at the lower limit from the value at the upper limit.

5. What are some real-life applications of integrating exponential functions?

Integrating exponential functions is used in various fields such as physics, engineering, and economics. It can be used to model population growth, radioactive decay, and compound interest, among others.

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