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# Classroom Tips and Techniques: Integration by Parts

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Classroom Tips and Techniques:

Integration by Parts in Maple

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

 Introduction Maple implements "integration by parts" with two different commands. One was designed in a pedagogical setting, and the other, for a "production" setting.   In this article, we compare the functionalities of the two commands, Parts  (in IntegrationTools)as a "production" tool, and Rule[parts]  (in Student:-Calculus1) as a didactic command. In the following illustrations, we will write the formula as   so that the factor  is be differentiated, and the companion factor, , is to be integrated.

Initialize

The following two integrals

and their sum

will be used as the basis for the discussion of parts integration.

In the second integral the textbook notation  stands for a Bessel function of order zero. Maple knows this function as "BesselJ" but the following modifications to Maple allow the textbook notation to be understood.

Alternatively, the interactive Rules Assistant accessed from the View menu can be used, provided extended typesetting has been implemented.

 Figure 1   Rules Assistant in default state Figure 2   Rules Assistant after parse rule enabled

The Parts Command from the IntegrationTools Package

Table 1 makes four applications of the Parts  command from the IntegrationTools package.

 Table 1   The Parts  command from the IntegrationTools package applied to the integrals in Table 1

Table 1 immediately reveals that only the term taken as u need be provided. The command then extracts dv and provides v. The table also reveals that the Parts  command is not deterred by a multiplicative factor in front of the integral. It also reveals that the command applies both to an expression as well as an equation. However, it further shows that if an inappropriate choice of u is made, a formal antiderivative representing v is devised, and the integration-by-parts formula is applied. Finally, the table reveals that if an expression contains more than one integral, an error results. The user must find a way to impose the command on just one integral in the expression.

Table 2 lists several ways that the Parts  command could be applied just to  in expression . Each requires that the integral to which Parts  is to be applied can be singled out by some distinguishing feature. In the first method, the evalindets  command is used to place the Parts  command onto the operand of  that is of Maple type `*`, that is, a product. The operands of  are identified by applying the op  command, as in

=

In the second method,  is decomposed by the alternate usage of the op  command, the Parts command is applied to the relevant integral (the first operand), and the transformed  reconstructed by adding back the second operant. Of course, this approach can be tedious, but that might be preferable to grappling with the mysteries of Maple types.

 Table 2   Applying Parts  to a particular integral in a sum of integrals

Rule[parts] from the Student[Calculus1] Package

The Student[Calculus1] package contains code for single-stepping through problems in calculus.  Limits, derivatives, and integrals can be evaluated step-wise, with the user signaling to Maple the relevant rules to apply.  Unfortunately, the syntax for this single-stepping by rules is more complicated than the typical instructor would want to expose to a class of calculus students.

Fortunately, this functionality has been embedded in pop-up tools that apply the rules via buttons in point-and-click interfaces called Tutors.  Figure 3 (below) shows the Integration Methods  tutor applied to . The first button to be pressed is the "Parts" button, which then launches the dialog in which both u and v are to be entered. Note that here, Maple uses the paradigm

 Figure 3   Stepwise integration by parts via the Integration Methods  tutor

In Table 3, the Rule[parts]  command is applied to the integrals examined in Table 1.

 Table 3    Rule[parts]  from the Student:-Calculus1 package applied to the integrals in Table 1

Unlike the Parts  command, Rule[parts]  requires the user to provide both u and v (not dv). But like the Parts  command in the IntegrationTools package, Rule[parts]  is not hampered by a multiplier in front of the integral. However, unlike the Parts  command, it is indeed impeded by an equation. It must act on an expression only. Finally, Rule[parts]  acts intelligently on an expression containing more than one integral.

 Conclusion The Parts  command in the IntegrationTools package maps across equations, but not across expressions containing more than one integral. Also, it acts formally on an integral for which the supplied  is inappropriate. But it need only be given u, not u and v. In the didactic environment, Rule[parts]  will remain the best alternative, mainly because it requires the student to provide  as well as , and behaves nicely in an equation and in the presence of more than one integral. Despite its shortcoming in an expression containing more than one integral, in a "production-mode" the Parts  command would be the tool of choice.

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