Overview - Maple Help

Overview of the IntegrationTools Package

 Calling Sequence IntegrationTools[command](arguments) command(arguments)

Description

 • The IntegrationTools package is a set of programmer tools used for low level manipulation of definite and indefinite integrals.
 Note: This package contains tools for manipulating the data structure only and do not ensure the validity of the operation being performed. For mathematical operations on integrals, use top-level commands such as combine, expand, etc., or the Student package.
 • At load time the IntegrationTools package defines three new types: Integral, DefiniteIntegral and IndefiniteIntegral, which can be used to access integrals involved in any given expression.
 • Each command in the IntegrationTools package can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 • As the underlying implementation of the IntegrationTools package is a module, it is also possible to use the form IntegrationTools:-command to access a command from the package. For more information,  see Module Members.

List of IntegrationTools Package Commands

 The following is a list of available commands.

 To display the help page for a particular IntegrationTools command, see Getting Help with a Command in a Package.

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $v≔\mathrm{Int}\left(f\left(x\right),x=a..b\right)$
 ${v}{≔}{{\int }}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (1)
 > $\mathrm{type}\left(v,\mathrm{Integral}\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{type}\left(v,\mathrm{DefiniteIntegral}\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{type}\left(v,\mathrm{IndefiniteIntegral}\right)$
 ${\mathrm{false}}$ (4)

Extract the integrand, variable of integration and range.

 > $\mathrm{GetIntegrand}\left(v\right)$
 ${f}{}\left({x}\right)$ (5)
 > $\mathrm{GetVariable}\left(v\right)$
 ${x}$ (6)
 > $\mathrm{GetRange}\left(v\right)$
 ${a}{..}{b}$ (7)

Split a definite integral.

 > $v≔\mathrm{Int}\left(\mathrm{sin}\left(x\right),x=0..2\mathrm{\pi }n\right)$
 ${v}{≔}{{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}{}{n}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (8)
 > $\mathrm{Split}\left(v,2\mathrm{\pi }\right)$
 ${{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{\int }}_{{2}{}{\mathrm{\pi }}}^{{2}{}{\mathrm{\pi }}{}{n}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (9)
 > $\mathrm{Split}\left(v,\left[2\mathrm{\pi },4\mathrm{\pi },6\mathrm{\pi }\right]\right)$
 ${{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{\int }}_{{2}{}{\mathrm{\pi }}}^{{4}{}{\mathrm{\pi }}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{\int }}_{{4}{}{\mathrm{\pi }}}^{{6}{}{\mathrm{\pi }}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{\int }}_{{6}{}{\mathrm{\pi }}}^{{2}{}{\mathrm{\pi }}{}{n}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (10)
 > $\mathrm{Split}\left(v,\left[2\mathrm{\pi }i,i=1..n-1\right]\right)$
 ${{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}\left({\sum }_{{\mathrm{_j}}{=}{1}}^{{n}{-}{2}}{}{{\int }}_{{2}{}{\mathrm{\pi }}{}{\mathrm{_j}}}^{{2}{}{\mathrm{\pi }}{}\left({\mathrm{_j}}{+}{1}\right)}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{{\int }}_{{2}{}{\mathrm{\pi }}{}\left({n}{-}{1}\right)}^{{2}{}{\mathrm{\pi }}{}{n}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (11)

Perform integration by parts.

 > $v≔\mathrm{Int}\left(\mathrm{exp}\left(x\right)\mathrm{sin}\left(x\right),x=a..b\right)$
 ${v}{≔}{{\int }}_{{a}}^{{b}}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (12)
 > $\mathrm{Parts}\left(v,\mathrm{sin}\left(x\right)\right)$
 ${{ⅇ}}^{{b}}{}{\mathrm{sin}}{}\left({b}\right){-}{{ⅇ}}^{{a}}{}{\mathrm{sin}}{}\left({a}\right){-}\left({{\int }}_{{a}}^{{b}}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (13)
 > $\mathrm{Parts}\left(v,\mathrm{exp}\left(x\right)\right)$
 ${-}{{ⅇ}}^{{b}}{}{\mathrm{cos}}{}\left({b}\right){+}{{ⅇ}}^{{a}}{}{\mathrm{cos}}{}\left({a}\right){-}\left({{\int }}_{{a}}^{{b}}{-}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (14)

Expand an integral.

 > $v≔\mathrm{Int}\left(af\left(x\right)+bg\left(x\right)+ch\left(x\right),x=1..2\right)$
 ${v}{≔}{{\int }}_{{1}}^{{2}}\left({a}{}{f}{}\left({x}\right){+}{b}{}{g}{}\left({x}\right){+}{c}{}{h}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (15)
 > $w≔\mathrm{Expand}\left(v\right)$
 ${w}{≔}{a}{}\left({{\int }}_{{1}}^{{2}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{b}{}\left({{\int }}_{{1}}^{{2}}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{c}{}\left({{\int }}_{{1}}^{{2}}{h}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (16)

Combine multiple integrals.

 > $\mathrm{Combine}\left(w\right)$
 ${{\int }}_{{1}}^{{2}}\left({a}{}{f}{}\left({x}\right){+}{b}{}{g}{}\left({x}\right){+}{c}{}{h}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (17)
 > $\mathrm{Combine}\left(\mathrm{Int}\left(f\left(x\right),x=a..b\right)+\mathrm{Int}\left(f\left(x\right),x=b..c\right)-\mathrm{Int}\left(f\left(x\right),x=a..d\right)\right)$
 ${{\int }}_{{d}}^{{c}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (18)