Integration Rules for Calculus1 - Maple Programming Help

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Integration Rules for Calculus1

Rules

 • See Student[Calculus1] for a general introduction to the Calculus1 subpackage of the Student package.
 • See SingleStepOverview for an introduction to the step-by-step (or single-step) functionality of the Calculus1 package.
 • The following table lists the built-in rules for integration that do not take parameters.  These rules can be passed as the index to Rule or as a rule argument to Understand.  These rules apply to both indefinite and definite integrals.

 Rule Alternate Names Description constant ${∫}c\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=cx$ constantmultiple $\mathrm{c*}$ ${∫}cf\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=c\left({∫}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$ diff $\mathrm{Diff}$ ${∫}\frac{{ⅆ}}{{ⅆ}x}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=f\left(x\right)$ difference $\mathrm{-}$ ${∫}\left(f\left(x\right)-g\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={∫}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x-\left({∫}g\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$ identity ${∫}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=\frac{1}{2}{x}^{2}$ partialfractions ${∫}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={∫}\left(R\left(x\right)+R\left(x\right)+\mathrm{...}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$ where $R\left(x\right)+R\left(x\right)+\mathrm{...}$ is a partial fraction decomposition of $f\left(x\right)$ power $\mathrm{^}$ ${∫}{x}^{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=\frac{{x}^{n+1}}{n+1}$ ${{∫}}_{a}^{b}{x}^{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=\frac{{b}^{n+1}}{n+1}-\frac{{a}^{n+1}}{n+1}$ revert revert a change of variables substitution solve algebraically solve an equation in which the same integral appears more than once (often used after multiple applications of integration by parts) sum $\mathrm{+}$ ${∫}\left(f\left(x\right)+g\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={∫}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x+{∫}g\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$

 The name of any univariate function can also be used as a rule argument to the Rule command.  The name of any univariate function recognized by Maple, for example, sin, can be passed as a rule argument to the Understand command (where recognized means that it is of type mathfunc).
 Note: Rules are named for properties or subexpressions of the integrand, not the result.  Thus, for example, the power rule, not the ln rule, applies to the problem of integrating $\frac{1}{x}$.
 • There are 3 additional rules that are recognized exclusively for definite integration: flip, join, and split.  Only split takes a parameter. None can be passed as a rule argument to Understand.

 Rule Description Parameters flip ${{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=-\left({{∫}}_{b}^{a}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$ join ${{∫}}_{a}^{c}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x+{{∫}}_{c}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$ split ${{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={{∫}}_{a}^{c}x\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x+{{∫}}_{c}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$ $c$

 • There are 3 additional integration rules that require one or more parameters: parts, rewrite, and change.  Each can be used as the index to Rule, but cannot be given as a rule argument to Understand.
 • The parts rule implements integration by parts: ${∫}f\left(x\right)\left(\frac{{ⅆ}}{{ⅆ}x}g\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=f\left(x\right)g\left(x\right)-\left({∫}g\left(x\right)\left(\frac{{ⅆ}}{{ⅆ}x}f\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$.  To apply the rule, you must specify the two functions $f\left(x\right)$ and $g\left(x\right)$.

 [parts, $f\left(x\right)$, $g\left(x\right)$]

 That is, the first parameter ($f\left(x\right)$) is the term to be differentiated, and the second parameter ($g\left(x\right)$) is the integral of the term to be integrated when applying the rule.
 Note: To determine the function $g\left(x\right)$ for integration by parts, set up a new problem. That is, if the integrand of the original problem is a product $f\left(x\right)G\left(x\right)$, you must determine $g\left(x\right)$ such that $g'\left(x\right)=G\left(x\right)$. Create the problem ${∫}G\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$, and then apply appropriate rules to find $g\left(x\right)$.  Use GetProblem to return to the original problem and apply integration by parts.
 • The rewrite rule changes the form of the integrand when the integration variable is not changed. This rule has the general form:

 [rewrite, $f\left(x\right)=g\left(x\right)$, $f\left(x\right)=g\left(x\right)$, ...]

 The effect of applying the rewrite rule is to perform each substitution listed as a parameter to the rule, where occurrences of the left-hand side of each substitution are replaced by the corresponding right-hand side.
 Note: The Rule routine does not attempt to validate the rewrite rules.
 • The change rule changes the integration variable.  It can take the following forms.

 [change, $F\left(x,u\right)=G\left(x,u\right)$, $u$, siderels] [change, $F\left(x,u\right)=G\left(x,u\right)$, $u=H\left(x\right)$, siderels]

 The first parameter ($F\left(x,u\right)=G\left(x,u\right)$) defines the relationship between the old variable ($x$) and the new variable ($u$).  This most commonly takes the form $x=G\left(u\right)$ or $u=F\left(x\right)$, but can be much more general, provided it can be solved for $x$.
 Note: The name of the new variable (u in the preceding paragraph) must be previously unused in the problem to which you are applying this change of variables rule.  If it is not new, the Rule routine automatically chooses a different name, and produces a corresponding userinfo (level 1) message.  The only exception to this uniqueness requirement occurs if the second parameter to the change rule is $x=u$, that is, the only effect of the change rule is to rename the variable of integration.  In this situation, if the new name is not new to the problem, a userinfo warning message is produced, but the renaming is performed. If it was not your intention to re-use a variable in the problem, you can invoke the Undo command to undo this change rule.
 The $\mathrm{siderels}$ parameter is always optional, and rarely needed; if provided, it must be the last parameter in the rule.  It can be used to provide some simplification rules (side relations) to be applied in the case that the Maple (automatic) simplification rules do not produce the desired form.  This parameter must be in the following form.

 $\left\{{f}_{1}\left(x\right)={g}_{1}\left(u\right),...,{h}_{1}\left(u\right)={k}_{1}\left(u\right),...\right\}$

 That is, it is a set of substitutions in which the old variable $x$ cannot appear on the right-hand side, and in which only one of the variables $x$ or $u$ can appear on the left-hand side.  Substitutions that depend on the old variable $x$ are applied before the main change of variables substitution, while substitutions that do not depend on $x$ are applied after.
 The remaining parameter ($u$ or $u=H\left(x\right)$) is also optional. It can be the new variable name ($u$) or can provide the formula for the inverse change of variables substitution ($u=H\left(x\right)$).  If the change of variables formula includes parameters it may be necessary to use one of these two forms to clearly specify the new variable.
 Note: If you provide the parameter $u=H\left(x\right)$ giving the inverse change of variables substitution, the Rule command does not attempt to verify its correctness.
 Note: When a change rule is applied to a definite integration problem, the Rule routine determines whether the end points of integration are transformed to the new coordinate system. The decision is based primarily on a measure of the simplicity of the transformed end points.  If the end points remain in terms of the original variable, end points appear in the form $x=a$, where $x$ is the original variable.  Once the integration has been completed, you can use the revert rule to transform back to the original variable, at which time the evaluation at the end points takes place.  Examples of these situations are given below.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Calculus1}\right):$
 > ${\mathrm{infolevel}}_{\mathrm{Student}[\mathrm{Calculus1}]}≔1:$
 > $\mathrm{Rule}[\mathrm{c*}]\left({{∫}}_{0}^{1}3\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$
 Creating problem #1
 ${{∫}}_{{0}}^{{1}}{3}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{3}{}\left({{∫}}_{{0}}^{{1}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (1)
 > $\mathrm{Understand}\left(\mathrm{Int},\mathrm{c*}\right)$
 ${\mathrm{Int}}{=}\left[{\mathrm{constantmultiple}}\right]$ (2)
 > $\mathrm{Rule}[\mathrm{parts},\mathrm{sin}\left(x\right),{ⅇ}^{x}]\left({∫}\mathrm{sin}\left(x\right){ⅇ}^{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$
 Creating problem #2
 ${∫}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}{-}\left({∫}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (3)
 > $\mathrm{Rule}[\mathrm{parts},\mathrm{cos}\left(x\right),{ⅇ}^{x}]\left(\right)$
 ${∫}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}{-}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right){-}\left({∫}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (4)
 > $\mathrm{Rule}[\mathrm{solve}]\left(\right)$
 ${∫}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({x}\right){}{{ⅇ}}^{{x}}{-}\frac{{1}}{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)$ (5)
 > $\mathrm{Rule}[\mathrm{change},u=2x]\left({{∫}}_{a}^{b}\mathrm{sin}\left(2x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$
 Creating problem #3 Applying substitution x = 1/2*u, u = 2*x with dx = 1/2*du, du = 2*dx
 ${{∫}}_{{a}}^{{b}}{\mathrm{sin}}{}\left({2}{}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}\frac{{1}}{{2}}{}{{∫}}_{{2}{}{a}}^{{2}{}{b}}{\mathrm{sin}}{}\left({u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}$ (6)
 > $\mathrm{Rule}[\mathrm{change},u=\sqrt{x}]\left({{∫}}_{a}^{b}\frac{\mathrm{sin}\left(\sqrt{x}\right)}{\sqrt{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$
 Creating problem #4 Applying substitution x = u^2, u = x^(1/2) with dx = 2*u*du, du = 1/2/x^(1/2)*dx
 ${{∫}}_{{a}}^{{b}}\frac{{\mathrm{sin}}{}\left(\sqrt{{x}}\right)}{\sqrt{{x}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{2}{}\left({{∫}}_{{x}{=}{a}}^{{x}{=}{b}}{\mathrm{sin}}{}\left({u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)$ (7)
 > $\mathrm{Rule}[\mathrm{sin}]\left(\right)$
 ${{∫}}_{{a}}^{{b}}\frac{{\mathrm{sin}}{}\left(\sqrt{{x}}\right)}{\sqrt{{x}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{2}{}\left(\genfrac{}{}{0}{}{\left({-}{\mathrm{cos}}{}\left({u}\right)\right)}{\phantom{{x}{=}{a}{..}{b}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\left({-}{\mathrm{cos}}{}\left({u}\right)\right)}}{{x}{=}{a}{..}{b}}\right)$ (8)
 > $\mathrm{Rule}[\mathrm{revert}]\left(\right)$
 Reverting substitution using u = x^(1/2)
 ${{∫}}_{{a}}^{{b}}\frac{{\mathrm{sin}}{}\left(\sqrt{{x}}\right)}{\sqrt{{x}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{-}{2}{}{\mathrm{cos}}{}\left(\sqrt{{b}}\right){+}{2}{}{\mathrm{cos}}{}\left(\sqrt{{a}}\right)$ (9)
 > $\mathrm{Rule}[\mathrm{split},c]\left({{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$
 Creating problem #5
 ${{∫}}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{{∫}}_{{a}}^{{c}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{∫}}_{{c}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (10)
 > $\mathrm{Rule}[\mathrm{flip}]\left(\right)$
 ${{∫}}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{-}\left({{∫}}_{{c}}^{{a}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{{∫}}_{{c}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (11)
 > $\mathrm{Rule}[\mathrm{join}]\left(\right)$
 ${{∫}}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{{∫}}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (12)