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

  • Thread starter hofer
  • Start date
  • Tags
    Integral
In summary, the conversation involves a person seeking help with integrating a complex polynomial using the gaussian error function and the possibility of using the multinomial theorem. It is suggested to use a numerical solution or a package like Maple or Mathematica. The conversation also includes a proposed solution involving derivatives and the multinomial theorem, but it is unsure if it will successfully represent the integral.
  • #1
hofer
3
0

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: [tex]\int_{0}^{1}{e^{(at + bt^2 + ct^3 + dt^4 + et^5 + ft^6 + gt^7)}dt}[/tex]
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 [tex]-\infty[/tex] and [tex]\infty[/tex].

Any help or clues about integrating it? Thank you.
 
Physics news on Phys.org
  • #2
You won't be able to integrate this one like you do with easier functions from textbooks.
 
  • #3
I see, thanks.

Any tip on how to solve it?
 
  • #4
Go for a numerical solution. A package like Maple of Mathematica can do it without any problem.
 
  • #5
Ok, thanks.
I will use gaussian quadrature of Mathematica.
 
  • #6
It is equal to the following derivatives:

[tex]
\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}
[/tex]
 
  • #7
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:

[tex]
\begin{align*}
\int_0^1 \exp(at+bt^2+ct^3+dt^4+et^5+ft^6+gt^7)dt&=
\int_0^1 \sum_{n=0}^{\infty} \frac{(at+bt^2+ct^3+dt^4+et^5+ft^6+gt^7)^n}{n!}\\
&=\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 \\
&=\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}
\end{align*}
[/tex]
 
Last edited:

1. What is the function of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7)?

The function e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) is an integral function, meaning it represents the total area under the curve of the given equation. It is often used in calculus to calculate the area under a curve when the function is continuously changing.

2. How do you solve for the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7)?

To solve for the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7), you can use integration techniques such as substitution or integration by parts. You can also use online calculators or software to calculate the integral for you.

3. What is the significance of the variables in the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7)?

The variable t represents the independent variable, or the input value, in the equation. The exponents of t (2, 3, 4, 5, 6, and 7) represent the different orders of the terms in the equation, which affect the shape of the curve and the rate at which it changes.

4. Can the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) be simplified?

Yes, the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) can be simplified using algebraic or trigonometric identities. However, the resulting integral may not always be in a simpler form and may require further integration techniques.

5. In what real-world applications is the integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) used?

The integral of e^(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) is used in many fields of science and engineering, particularly in physics, chemistry, and economics. It is used to calculate the total change in a quantity over time, such as velocity, acceleration, or growth rate. It is also used in optimization problems to find the maximum or minimum value of a function.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
531
  • Calculus and Beyond Homework Help
Replies
2
Views
186
  • Calculus and Beyond Homework Help
Replies
3
Views
181
  • Topology and Analysis
Replies
4
Views
1K
Replies
6
Views
857
  • Calculus and Beyond Homework Help
Replies
17
Views
3K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
Replies
9
Views
3K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Back
Top