Integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7)

[hofer]

Homework Statement

After some work, I have come across an integral, and I have no idea how to develop it.

Homework Equations

The integral is: $$\int_{0}^{1}{e^{(at + bt^2 + ct^3 + dt^4 + et^5 + ft^6 + gt^7)}dt}$$
a, b, c, d, e, f, g are real constants.

The Attempt at a Solution

A friend told me to look into the gaussian error function, but I don't think it applies here. A wikipedia entry about Higher-order polynomials (http://en.wikipedia.org/wiki/Gaussian_integral#Higher-order_polynomials) says that it could be integrated using series, but only if the integration interval is between $$-\infty$$ and $$\infty$$.

Any help or clues about integrating it? Thank you.

 

[Outlined]

You won't be able to integrate this one like you do with easier functions from textbooks.

 

[hofer]

I see, thanks.

Any tip on how to solve it?

 

[Outlined]

Go for a numerical solution. A package like Maple of Mathematica can do it without any problem.

 

[hofer]

Ok, thanks.
I will use gaussian quadrature of Mathematica.

 

[Dickfore]

It is equal to the following derivatives:

$$\exp\left(b \frac{d^2}{d a^{2}} + c \, \frac{d^{3}}{d a^{3}} + d \, \frac{d^{4}}{d a^{4}} + e \, \frac{d^{5}}{d a^{5}} + f \, \frac{d^{6}}{d a^{6}} + g \, \frac{d^{7}}{d a^{7}} \right) \, \frac{\exp(a) - 1}{a}$$

 

[jackmell]

I'm not sure if the multinomial theorem could successfully represent the value of this integral so I'll propose it only as a possibility:

$$\begin{align*}<br /> \int_0^1 \exp(at+bt^2+ct^3+dt^4+et^5+ft^6+gt^7)dt&=<br /> \int_0^1 \sum_{n=0}^{\infty} \frac{(at+bt^2+ct^3+dt^4+et^5+ft^6+gt^7)^n}{n!}\\<br /> &=\int_0^1 \sum_{n=0}^{\infty}\frac{1}{n!}\sum_{k_1,k_2\cdots,k_7}\binom{n}{k_1,k_2,\cdots,k_7} (at)^{k_1}(bt^2)^{k_2}\cdots (gt^7)^{k_7} dt \\<br /> &=\sum_{n=0}^{\infty}\sum_{k_1,k_2,\cdots,k_7}\binom{n}{k_1,k_2,\cdots,k_7}\frac{a^{k_1}b^{2k_2}\cdots g^{7k_7}}{k_1+2k_2+\cdots 7k_7+1}<br /> \end{align*}$$