Integration Methods - Maple Programming Help

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Integration Methods

Definite Integration Options

Description

 The following integration methods can be specified with the method option to int.
 • method=_DEFAULT forces use of the default integration method.  It runs all of the integrators in sequence and returns the first answer found.
 • method=_UNEVAL causes the integrator to return unevaluated without trying any integration methods.
 • method=Integrator runs only the named Integrator and returns the result or unevaluated.  The integrator names are not case sensitive. The most interesting integrators for users are:
 – LookUp tries to find the integral in a lookup table.
 – FTOC applies the fundamental theorem of calculus using indefinite integration and limits. The method FTOCMS does the same, but uses the limit implementation in MultiSeries.
 – Elliptic applies methods to rewrite an integral in terms of elliptic integrals.  See the elliptic_int help page. There is also an EllipticTrig method which applies substitutions to find trig and hyperbolic trig forms of elliptic integrals.
 – Polynomial directly computes the integral algebraically if it is a polynomial.  Ratpoly does the same with rational functions.
 – MeijerG attempts to integrate by converting the integrand into an expression in terms of MeijerG functions.
 • Running int with infolevel[IntegrationTools] set to 3 will show the list of integrators run.
 • method=NoIntegrator runs the default integration method but skips the any integrator with a name prefixed by Integrator. e.g. NoElliptic skips methods Elliptic and EllipticTrig.
 • method=NoXXIntegrator skips only the named integrator.  e.g. NoXXElliptic skips only method Elliptic.
 • method=[method1, method2, etc] combines methods.  If the methods are integrators, then each is tried in sequence.  If the methods are all of the form NoIntegrator then they are each removed from the default integration sequence.  A list with one method or a list combining Integrator and NoIntegrator methods is not particularly useful, but both are supported.  _UNEVAL overrides any other methods it might be combined with and _DEFAULT is overridden by any other methods.

Examples

 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{FTOC}\right)$
 ${\mathrm{EllipticK}}{}\left(\sqrt{{2}}\right)$ (1)
 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{Elliptic}\right)$
 ${-}\frac{{I}{}\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}{+}\frac{\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}$ (2)
 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{NoElliptic}\right)$
 ${\mathrm{EllipticK}}{}\left(\sqrt{{2}}\right)$ (3)

If a method does not return a result, an unevaluated int call will be returned with the given method option.

 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{Polynomial}\right)$
 ${\mathrm{int}}{}\left(\frac{{1}}{\sqrt{\left({-}{{t}}^{{2}}{+}{1}\right){}\left({-}{2}{}{{t}}^{{2}}{+}{1}\right)}}{,}{t}{=}{0}{..}{1}{,}{\mathrm{method}}{=}{\mathrm{Polynomial}}\right)$ (4)
 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{_UNEVAL}\right)$
 ${\mathrm{int}}{}\left(\frac{{1}}{\sqrt{\left({-}{{t}}^{{2}}{+}{1}\right){}\left({-}{2}{}{{t}}^{{2}}{+}{1}\right)}}{,}{t}{=}{0}{..}{1}{,}{\mathrm{method}}{=}{\mathrm{_UNEVAL}}\right)$ (5)
 > $\mathrm{infolevel}\left[\mathrm{IntegrationTools}\right]≔3:$
 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(\left(1-{t}^{2}\right)\left(1-2{t}^{2}\right)\right)},t=0..1,\mathrm{method}=\mathrm{_DEFAULT}\right)$
 Definite Integration:   Integrating expression on t=0..1 Definite Integration:   Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour] LookUp Integrator:   unable to find the specified integral in the table Definite Integration:   Method elliptic succeeded. Definite Integration:   Finished sucessfully.
 ${-}\frac{{I}{}\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}{+}\frac{\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)}{{2}}$ (6)