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LayMuon

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[itex]

\int_0^\pi \exp^{-a x^2 -b x^4} dx \\

\int_0^\pi \exp^{-a x^2 -b x^4} (x - x^3)dx

[/itex]

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- Thread starter LayMuon
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In summary, the integral of exponential with polynomial argument is more complex than a regular exponential integral due to the addition of a polynomial argument. It can be solved using substitution, but may require multiple substitutions and can be a lengthy process. There are special cases where the integral can be solved using the regular exponential integral formula or substitution for a linear function. Some real-world applications of this integral include modeling population growth, radioactive decay, and analyzing systems in physics, engineering, and economics.

- #1

LayMuon

- 149

- 1

[itex]

\int_0^\pi \exp^{-a x^2 -b x^4} dx \\

\int_0^\pi \exp^{-a x^2 -b x^4} (x - x^3)dx

[/itex]

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Wolfram Alpha.

- #3

LayMuon

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Doesn't evaluate the first one.

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The formula for finding the integral of exponential with polynomial argument is:

∫ e^ax^n dx = (e^ax^n)/(a(n+1)) + C

The difference between the integral of exponential with polynomial argument and a regular exponential integral is that the polynomial argument adds an additional variable to the equation. This means that the power of the exponential term is not a constant, but rather a polynomial expression. This makes the integration more complex compared to a regular exponential integral.

Yes, the integral of exponential with polynomial argument can be solved using substitution. However, it may require multiple substitutions and can be a more lengthy process compared to other integration techniques such as integration by parts or partial fractions.

Yes, there are certain special cases for solving the integral of exponential with polynomial argument. For example, if the polynomial argument is a constant, then the integral can be solved using the regular exponential integral formula. Another special case is when the polynomial argument has a degree of 1, making it a linear function, which can also be solved using substitution.

The integral of exponential with polynomial argument has various real-world applications in fields such as physics, engineering, and economics. For example, it can be used to model the growth of populations, the decay of radioactive substances, and the change in value of investments over time. It is also used in signal processing and control systems to analyze and predict the behavior of systems over time.

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