Integrals of Exponential(Polynomial(x)) dx Form

In summary, the conversation discusses the general solution to the integral of a polynomial with real coefficients and a negative leading order coefficient. The participants mention having trouble calculating the integral for quartic equations and mention looking for a reference with a good exposition of the solution.
  • #1
RDBaker
4
0
I'm curious about the general solution to

[itex]\int_{-\infty}^{+\infty} \exp[P(x)] dx [/itex]

Where P(x) is a polynomial in x with real coefficients and whose leading (highest) order is even and its leading order coefficient is negative. Intuitively these integrals ought to converge, but I'm having trouble calculating them.

I've been able to work out solutions for quadratics i.e. P(x) = -ax^2 +bx +c, but I'm thoroughly stuck w.r.t. quartic equations.

Has anyone ever seen anything like this? Gradshteyn & Rizhyk and mathematica have been of no use to me.
 
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  • #3
Thanks! This is great!

Next step is trying to find a reference and a text with a good exposition of this solution.
 
  • #4
Hi RDBaker,

I am actually interested in exactly the same problem but have had trouble finding a more comprehensive reference. Did you have any luck?

Thanks a lot!
 

Related to Integrals of Exponential(Polynomial(x)) dx Form

1. What is the purpose of finding the integral of an exponential polynomial?

The purpose of finding the integral of an exponential polynomial is to determine the area under the curve of the function. This can be useful in various applications, such as calculating probabilities in statistics or determining the total amount of a substance in a chemical reaction.

2. How do you solve an integral of an exponential polynomial?

To solve an integral of an exponential polynomial, you can use various techniques such as integration by parts, substitution, or partial fraction decomposition. It is important to first identify the form of the exponential polynomial and then choose the appropriate integration method.

3. Can integrals of exponential polynomials have different forms?

Yes, integrals of exponential polynomials can have different forms depending on the structure of the function. Some common forms include exponential, logarithmic, trigonometric, and hyperbolic functions. Each form requires a different approach to solve the integral.

4. Is there a general formula for finding the integral of an exponential polynomial?

No, there is no general formula for finding the integral of an exponential polynomial. The method used to solve the integral depends on the specific form of the function. However, there are some common techniques and rules that can be applied depending on the form.

5. Can integrals of exponential polynomials be solved using software or calculators?

Yes, most software and calculators have built-in functions for solving integrals, including those of exponential polynomials. However, it is still important to understand the concepts and techniques behind solving integrals in order to use these tools effectively and accurately.

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