# Integration of exponential function times polynomial of fractional degree

• andyyee
In summary: L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}.- However, he has no idea how to solve the integral and gets help from the forums.- He finds that \Gamma(3/2) is just a constant and that there is a relationship between the Gamma value at 3/2 and 1/2.
andyyee

## Homework Statement

I'm working out a differential equation problem that I am supposed to solve with the formula $\mathcal{L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}$. The problem is $\mathcal{L}\{t^{\frac{1}{2}}\}$ (finding the Laplace transform of the given function).

## Homework Equations

$\mathcal{L}\{t^\alpha\} = \frac{\Gamma(\alpha + 1)}{s^{\alpha+1}}, \alpha > -1$

$\Gamma(\alpha) = \int^\infty_0{t^{\alpha-1}e^{-t}dt}, \alpha > 0$

## The Attempt at a Solution

I plug it into the equation, and get $\frac{\Gamma(\frac{3}{2})}{s^\frac{3}{2}} =$ $\frac{\int^\infty_0 {t^\frac{1}{2}e^{-t}dt}}{s^{3/2}}$. That's where I run into a problem, I have no idea how to solve that integral. I can't use integration by parts, because one term will never disappear or the original integral will not appear again as $\int{vdu}$, so that won't work. I looked it up on Wolfram|Alpha, and it gave me $\frac{1}{2}\sqrt{\pi}\text{erf}{(\sqrt{t})} - e^{-t}\sqrt{t}$ for the indefinite form and $\frac{\sqrt{\pi}}{2}$ for the definite form. It also cited some stuff about integrating the normal distribution and error form, but I don't understand what it is talking about. What I am unsure about is the steps involved in solving the integral, and is there a generalized solution (for the definite integral from 0 to $\infty$) for other coefficient values in the power of the polynomial?

Thanks
Andrew

But you don't have to do the integral. You have the LaPlace transform from your formula;

$$\frac{\Gamma(\frac 3 2)}{s^\frac 3 2}$$

$\Gamma(3/2)$ is just a constant. Look in your text and see if it doesn't give you $\Gamma(1/2)$ and a relationship between the Gamma value at 3/2 and 1/2.

Thanks for the help :-). I completely overlooked the part in my book that mentioned an appendix, which happened to show some basic integration (for gamma of 1/2 using double integrals), as well as ways to solve similar gamma values without integration (simple algebra transformations using the already computed, given gamma of 1/2 and some other formulas). These forums are really great, because although Wolfram|Alpha gives me solutions, it does not tell me why (or to look closer at my book :-).

- Andrew

## What is the Integration of Exponential Function Times Polynomial of Fractional Degree?

The integration of exponential function times polynomial of fractional degree is a mathematical process that involves finding the antiderivative of a function that is the product of an exponential function and a polynomial of fractional degree. It is often used in calculus and engineering to solve problems involving growth and decay.

## What are the Steps for Integrating Exponential Function Times Polynomial of Fractional Degree?

The steps for integrating exponential function times polynomial of fractional degree include using integration by parts, substitution, and partial fractions. The specific steps may vary depending on the complexity of the function, but the general process involves breaking down the function into simpler parts and applying integration techniques to each part.

## What are the Applications of Integrating Exponential Function Times Polynomial of Fractional Degree?

The integration of exponential function times polynomial of fractional degree has many real-world applications, such as in finance, physics, and chemistry. It is used to model various phenomena, such as population growth, radioactive decay, and chemical reactions. It is also essential in solving differential equations, which are used to describe many natural processes.

## What are the Challenges in Integrating Exponential Function Times Polynomial of Fractional Degree?

Integrating exponential function times polynomial of fractional degree can be challenging due to the complexity of the functions involved. It requires a good understanding of various integration techniques and the ability to manipulate and simplify equations. The presence of fractions and negative exponents can also make the integration process more challenging.

## Are There Any Strategies for Simplifying the Integration of Exponential Function Times Polynomial of Fractional Degree?

Yes, there are some strategies that can make the integration of exponential function times polynomial of fractional degree easier. These include factoring, substitution, and using trigonometric identities. It is also helpful to break down the function into smaller parts and apply integration techniques to each part separately. Practice and familiarity with integration methods can also help simplify the process.

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