2026-05-28T23:42:09Zhttps://riuma.uma.es/rest/oai/requestoai:riuma.uma.es:10630/78502026-02-03T12:18:03Zcom_10630_2254col_10630_37959
Advanced techniques to compute improper integrals using a CAS
Aguilera-Venegas, Gabriel
Galán-García, José Luis
Galán-García, María Ángeles
Padilla-Domínguez, Yolanda Carmen
Rodríguez-Cielos, Pedro
Rodríguez-Cielos, Ricardo
Fourier, Transformaciones de
Laplace, Transformación de
Let us consider the following types of improper integrals:
$$
\int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t
$$
\medskip
Let $F$ be an antiderivative of $f$. The basic approach to compute such integrals involves the following computations:
\medskip
\begin{eqnarray*}
\int_0^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_0^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(0)\big) \\ \\
\int_{-\infty}^0 f(t)\:{\rm d}t & = & \lim_{m\to-\infty} \int_m^0 f(t)\:{\rm d}t = \lim_{m\to-\infty} \big(F(0)-F(m)\big) \\ \\
\int_{-\infty}^\infty f(t)\:{\rm d}t & = & \int_{-\infty}^0 f(t)\:{\rm d}t + \int_0^\infty f(t)\:{\rm d}t \qquad \mbox{or, in case of convergence,} \\ \\
\int_{-\infty}^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_{-m}^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(-m)\big) \qquad \mbox{(Cauchy principal value)}
\end{eqnarray*}
\medskip
\noindent But, what happens if an antiderivative $F$ for $f$ or the above limits do not exist?
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\noindent For example, for \quad $\displaystyle\int_0^\infty\frac{{\rm sin}(at)}{t}\:{\rm d}t$ \quad ; \quad $\displaystyle\int_0^\infty\frac{{\rm cos}(at)-{\rm cos}(bt)}{t}\:{\rm d}t$ \quad {\rm or} \quad $\displaystyle\int_{-\infty}^\infty\frac{{\rm cos}(at)}{t^2+1}\:{\rm d}t$ \qquad the antiderivatives can not be computed. Hence, the above procedures cannot be used for these examples.
\medskip
In this work we will deal with advance techniques to compute this kind of improper integrals using a {\sc Cas}. Laplace and Fourier transforms or Residue Theorem in Complex Analysis are some advance techniques which can be used for this matter.
\medskip
We will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations.
\medskip
Some {\sc Cas} use different rules for computing integrations. For example {\sc Rubi} system, a {\bf ru}le-{\bf b}ased {\bf i}ntegrator developed by Albert Rich (see {\tt http://www.apmaths.uwo.ca/\~{ }arich/}), is a very powerful system for computing integrals using rules. We will be able to develop new rules schemes for some improper integrals using {\tt ImproperIntegrals.mth}. These new rules can extend the types of improper integrals that a {\sc Cas} can compute.
2014-07-16T10:47:39Z
2014-07-16T10:47:39Z
2014-07-16T10:47:39Z
2014-07-16
conference output
http://hdl.handle.net/10630/7850
eng
Technology and its Integration in Mathematics Education TIME 2014
Krems, Austria
1-5/07/2014
open access