1.
(a)
Plot \(r(x)\text{.}\)
(b)
Evaluate the integral \(\displaystyle\int_{-\infty}^{-1} r(x) dx\text{.}\)
Section 3.11 Infinite Integrals
Subsection 3.11.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 3.11.2 Introduction
Infinite integrals are used in a variety of applications, including finding solutions to differential equations by way of the Laplace transform. Infinite integrals can be challenging to evaluate by hand, but in this activity you will see that Maple can handle infinite integrals easily.
Exercises 3.11.3 Exercises
2.
Assign the function \(g(x) = x{ e}^{-x^2}\) in Maple.
(a)
Plot \(g(x)\text{.}\)
(b)
Evaluate the integral \(\displaystyle\int_{-\infty}^{\infty} g(x) dx\text{.}\)
3.
Assign the function \(h(x) = \dfrac{\ln(x)}{x}\) in Maple.
(a)
Plot \(h(x)\text{.}\)
(b)
Evaluate the integral \(\displaystyle\int_{1}^{\infty} h(x) dx\text{.}\)
4.
The Laplace transform of a function \(f(t)\) is an integral operation given by
\begin{equation}
\mathcal{L}\lbrace f \rbrace(s) = \displaystyle\int_{0}^{\infty} f(t){ e}^{-st} dt\text{.}\tag{3.8}
\end{equation}
Note that this integral involves the integrating variable, \(t\text{,}\) as well as a second variable, \(s\text{.}\) The result after the definite integral is a function of \(s\text{,}\) assuming that the integral is convergent.
(a)
You will need to begin by telling Maple to assume that \(s\) is a positive real number (as opposed to a complex number). To do this, run the following command on a new line:
> assume(s, positive);
Maple will now assume that \(s\) can only be a real-valued variable. However, it defaults to showing this assumption by outputting a tilde character \(\sim\) whenever the variable shows up in output. You can hide assumptions on variables by running a second command:
> interface(showassumed=0);
Run both of these commands so that you can evaluate the Laplace transform in part (b).
