Question

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $${\int }_0^{\mathrm{\infty }}{x}^{5/2}{e}^{-x}dx$$

Short answer

The integral evaluates to $\frac{15\sqrt{\pi }}{4}$.

Step-by-step solution

- Identify the integral form

The given integral is ${\int }_0^{\mathrm{\infty }}{x}^{5/2}{e}^{-x}dx$. Recognize that this integral matches the form of the Gamma function integral, ${\int }_0^{\mathrm{\infty }}{x}^{n-1}{e}^{-x}dx=\mathrm{\Gamma }(n)$.

- Express the exponent in Gamma function form

Rewrite the integral such that the exponent in the integrand matches the format of the Gamma function: $n-1=\frac{5}{2}$. Therefore, $n=\frac{7}{2}$.

- Use the Gamma function definition

Using $=\frac{7}{2}$, apply the Gamma function definition: ${\int }_0^{\mathrm{\infty }}{x}^{\frac{5}{2}}{e}^{-x}dx=\mathrm{\Gamma }(\frac{7}{2})$.

- Evaluate the Gamma function

The value of the Gamma function for $\frac{7}{2}$ is known: $\mathrm{\Gamma }(\frac{7}{2})=\frac{5}{2}\cdot \frac{3}{2}\cdot \mathrm{\Gamma }(\frac{1}{2})$. Since $\mathrm{\Gamma }(\frac{1}{2})=\sqrt{\pi }$, the result is: $\mathrm{\Gamma }(\frac{7}{2})=\frac{15\sqrt{\pi }}{4}$.

- Compare with computer evaluation

Check the evaluated result using a computer algebra system: ${\int }_0^{\mathrm{\infty }}{x}^{5/2}{e}^{-x}dx=\frac{15\sqrt{\pi }}{4}$. Confirm the answer matches.

Key concepts

Definite integrals

Definite integrals are a core concept in calculus that evaluate the accumulation of quantities, typically over a specified interval. In this exercise, we handle a definite integral with its bounds from 0 to infinity. This integral provides a sum of infinite small pieces. To get the same result as the original problem, it's crucial to understand this structure.
When working with definite integrals:

• Identify the limits of integration, which tell you the interval over which you’re integrating.
• Understand the integrand, the function you are integrating, as it dictates how the accumulation happens.
• Use properties and techniques such as substitution or recognizing standard forms to simplify and evaluate the integral.

The integral given in the exercise, $\begin{array}{r}{\int }_0^{\mathrm{\infty }}{x}^{5/2}{e}^{-x}dx\end{array}$, fits the structure of the Gamma function and evaluates the accumulation of the expression from 0 to infinity.

Gamma function properties

The Gamma function, $\mathrm{\Gamma }(n)$, generalizes the factorial function to real and complex numbers. It's used in many areas, including probability and statistics.
One key property of the Gamma function is its integral representation:
$$\mathrm{\Gamma }(n)={\int }_0^{\mathrm{\infty }}{x}^{n-1}{e}^{-x}dx$$ This matches our problem well, allowing us to rewrite the given integral in terms of $\mathrm{\Gamma }$.
To use this form, notice that:
$n-1=\frac{5}{2}\Rightarrow n=\frac{7}{2}$.
Other useful properties include:

• $\mathrm{\Gamma }(n+1)=n\cdot \mathrm{\Gamma }(n)$
• $\mathrm{\Gamma }(1/2)=\sqrt{\pi }$

Using these properties, we can simplify and evaluate $\mathrm{\Gamma }(7/2)$ in the solution as:
$\mathrm{\Gamma }(7/2)=\frac{5}{2}\cdot \frac{3}{2}\cdot \mathrm{\Gamma }(1/2)\to \frac{15\sqrt{\pi }}{4}$.

Evaluation techniques

When solving integrals, various techniques can help you efficiently reach the solution. Recognizing the form of the integral, as done in this exercise, is crucial.
Here’s how you can approach it efficiently:

• Identify if the integral fits a known form, like the Gamma function in this case.
• Transform the integral, aligning it with the known properties. Here, that involves rewriting $n-1$ in Gamma’s integral form.
• Apply the relevant properties and rules, such as breaking down $\mathrm{\Gamma }(7/2)$ using multiplication properties.

In our exercise, these steps brought us to find $\mathrm{\Gamma }(7/2)$ directly. Evaluating $\mathrm{\Gamma }(7/2)$ using properties of Gamma made processing the integral straightforward.
Make use of algebraic substitutions or known values, such as $\mathrm{\Gamma }(1/2)$, to reach the final result: $\frac{15\sqrt{\pi }}{4}$. Techniques like these are foundational for more advanced calculus problems.