The following options, which are described in more detail in the sections below, apply to definite integrals only. Where indicated, some options take their default value from environment variables. An option has a default value of false if its corresponding environment variable is not set.

AllSolutions=truefalse

continuous=truefalse

CauchyPrincipalValue=truefalse

method=value 

numeric=truefalse

useunits=truefalse or useunits=list of unit specifications

This help page contains complete information about the int and Int commands. For basic information, see the int help page.

The int command computes an indefinite or definite integral of the expression expression with respect to the variable x. The name integrate is a synonym for int.

You can enter the command int using either the 1-D or 2-D calling sequence.  For instance, int(f,x) is equivalent to $\int f\ⅆx$.

Indefinite integration is performed if the second argument x is a name. Note that no constant of integration appears in the result. Definite integration is performed if the second argument is of the form x=a..b where a and b are the endpoints of the interval of integration.

If a and b are finite complex numbers, the int routine computes the definite integral over the straight line from a to b.

If the integrand is specified as a Maple operator f, then the second argument must be a range a..b and not an equation, that is, a variable of integration must not be specified.

If Maple cannot find a closed form expression for the integral, the function call itself is returned.

The capitalized function name Int is the inert version of the int function, which simply returns unevaluated. It appears gray so that it is easily distinguished from a returned int calling sequence. In this form, expression can be a procedure, which can be integrated numerically.

For numerical integration, use the option numeric.  For more information on numerical integration, see evalf/Int.

If the second argument is a list, an iterated integration is performed on the variables or ranges in the order given. For definite integration, the list can be omitted and the ranges can be given as a sequence. For indefinite integration, the variables can be given as a sequence instead of a list if all variables appear in the expression being integrated. Otherwise, there may be confusion between variables and option names.

The Maple series function may be invoked on an unevaluated integral to compute a series expansion of the integral (when possible).

When int is applied to a series structure, the internal function `int/series` is invoked to compute the integral efficiently.

$\mathrm{int}\left(\sin \left(x\right)\,x\right)$

$-\cos \left(x\right)$

$\mathrm{int}\left(\sin \left(x\right)\,x=0..\mathrm{\pi }\right)$

$2$

$\mathrm{int}\left(\sin \,a..b\right)$

$\cos \left(a\right)-\cos \left(b\right)$

$\mathrm{int}\left(\frac{x}{x^3-1}\,x\right)$

$\frac{\ln \left(x-1\right)}{3}-\frac{\ln \left(x^2+x+1\right)}{6}+\frac{\sqrt{3}\arctan \left(\frac{\left(2x+1\right)\sqrt{3}}{3}\right)}{3}$

$\mathrm{int}\left(\exp \left(-x^2\right)\,x\right)$

$\frac{\sqrt{\mathrm{\pi }}\mathrm{erf}\left(x\right)}{2}$

$\mathrm{int}\left(\exp \left(-x^2\right)\ln \left(x\right)\,x=0..\mathrm{\infty }\right)$

$-\frac{\sqrt{\mathrm{\pi }}\mathrm{\gamma }}{4}-\frac{\sqrt{\mathrm{\pi }}\ln \left(2\right)}{2}$

$\mathrm{int}\left(\exp \left(-x^2\right)\ln \left(x\right)\,x\right)$

$\int {\ⅇ}^{-x^2}\ln \left(x\right)\ⅆx$

$\mathrm{series}\left(\,x=0\,4\right)$

$\left(\ln \left(x\right)-1\right)x+\left(-\frac{\ln \left(x\right)}{3}+\frac{1}{9}\right)x^3+O\left(x^5\right)$

$\mathrm{int}\left(\,x\right)$

$\frac{\ln \left(x\right)x^2}{2}-\frac{3x^2}{4}-\frac{x^4\ln \left(x\right)}{12}+\frac{7x^4}{144}+\mathrm{O}\left(x^6\right)$

$\mathrm{int}\left(\exp \left(-x^2\right){\ln \left(x\right)}^2\,x=0..\mathrm{\infty }\right)$

$\frac{{\mathrm{\pi }}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{16}+\frac{{\mathrm{\gamma }}^2\sqrt{\mathrm{\pi }}}{8}+\frac{\mathrm{\gamma }\sqrt{\mathrm{\pi }}\ln \left(2\right)}{2}+\frac{\sqrt{\mathrm{\pi }}{\ln \left(2\right)}^2}{2}$

$f≔7x^3+3x^2+5x\:$

$\mathrm{Int}\left(f\,x\right)$

$\int \left(7x^3+3x^2+5x\right)\ⅆx$

$\mathrm{value}\left(\right)$

$\frac{7}{4}{x}^4+x^3+\frac{5}{2}{x}^2$

$\mathrm{Int}\left(x{y}^2\,\left[x=0..1\,y=0..1\right]\right)$

${\int }_0^1{\int }_0^{1}x{y}^2\ⅆx\ⅆy$

$\mathrm{int}\left(x{y}^2\,x\,y\right)$

$\frac{x^2{y}^3}{6}$

$\mathrm{int}\left(x{y}^2\,\left[x=0..y\,y=-2..2\right]\right)$

$\frac{32}{5}$

$\mathrm{int}\left(\left(t^2-1\mathrm{Unit}\left(s^2\right)\right)\mathrm{Unit}\left(\frac{m}{s^3}\right)\,t=0..2\mathrm{Unit}\left(s\right)\right)$

$\frac{2}{3}⟦m⟧$

$\mathrm{int}\left(\left(t^2-1\mathrm{Unit}\left(s^2\right)\right)\mathrm{Unit}\left(\frac{m}{s^3}\right)\,t\,\mathrm{useunits}=\left[t::\mathrm{Unit}\left(s\right)\right]\right)$

$\frac{t^3}{3}⟦\frac{m}{s^3}⟧-t⟦\frac{m}{s}⟧$

$\mathrm{int}\left(\left(t^2-1\mathrm{Unit}\left(s^2\right)\right)\mathrm{Unit}\left(\frac{m}{s^3}\right)\,t\,\mathrm{useunits}=\left[t::\mathrm{Unit}\left(m\right)\right]\right)$

Error, (in Tools:-IntegrateExpression) units problem can't make expression (t^2-Units:-Unit(s^2))*Units:-Unit(m/s^3) unit-free with units [t::Units:-Unit(m)]

$\mathrm{int}\left(t\mapsto \left(t^2-\mathrm{Unit}\left(h^2\right)\right)\cdot \mathrm{Unit}\left(\frac{m}{h^3}\right)\,1.\mathrm{Unit}\left(s\right)..2.\mathrm{Unit}\left(s\right)\,\mathrm{numeric}\,\mathrm{useunits}=\left[\mathrm{Unit}\left(h\right)\right]\right)$

$-0.0002777777278⟦m⟧$

$\mathrm{int}\left(\left(t^2-1\mathrm{Unit}\left(h^2\right)\right)\mathrm{Unit}\left(\frac{m}{h^3}\right)\,t=1\mathrm{Unit}\left(\mathrm{ft}\right)..2\mathrm{Unit}\left(m\right)\right)$

Error, (in Tools:-IntegrateExpression) units problem can't make expression (t^2-Units:-Unit(h^2))*Units:-Unit(m/h^3) unit-free with units [t::Units:-Unit(ft)]

$\mathrm{int}\left(\left(t^2-1\mathrm{Unit}\left(h^2\right)\right)\mathrm{Unit}\left(\frac{m}{h^3}\right)\,t=1\mathrm{Unit}\left(\mathrm{ft}\right)..2\mathrm{Unit}\left(m\right)\,\mathrm{useunits}=\mathrm{false}\right)$

$\frac{15569693659}{5859375000}⟦\frac{m^4}{h^3}⟧-\frac{2119}{1250}⟦\frac{m^2}{h}⟧$

$\mathrm{int}\left(\frac{1}{{\left(x+a\right)}^2}\,x=0..2\,'\mathrm{continuous}'\right)$

$\frac{2}{a\left(2+a\right)}$

$\mathrm{int}\left(\frac{1}{x^3}\,x=-1..2\,'\mathrm{CauchyPrincipalValue}'\right)$

$\frac{3}{8}$

$\mathrm{int}\left(F\left(x\right)\mathrm{Heaviside}\left(x-1\right)\,x=-1..3\right)$

${\int }_1^{3}F\left(x\right)\ⅆx$

$\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(2t^4-3t^2-2\right)}\,t=2..3\right)$

$\frac{\sqrt{5}\mathrm{EllipticF}\left(\frac{\sqrt{7}}{3}\,\frac{\sqrt{5}}{5}\right)}{5}-\frac{\sqrt{5}\mathrm{EllipticF}\left(\frac{\sqrt{2}}{2}\,\frac{\sqrt{5}}{5}\right)}{5}$

$f≔\frac{\frac{x^2-1+3{\left(1+x^2\right)}^{\frac{1}{3}}}{{\left(1+x^2\right)}^{\frac{2}{3}}}x}{{\left(x^2+2\right)}^2}$

$f≔\frac{\left(x^2-1+3{\left(x^2+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)x}{{\left(x^2+1\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(x^2+2\right)}^2}$

$f≔\mathrm{convert}\left(f\,\mathrm{RootOf}\right)\:$

$g≔\mathrm{int}\left(f\,x\right)$

$g≔-\frac{3\mathrm{RootOf}\left({\mathrm{\_Z}}^3-x^2-1\,\mathrm{index}=1\right)}{2\left(x^2+2\right)}+\frac{3{\mathrm{RootOf}\left({\mathrm{\_Z}}^3-x^2-1\,\mathrm{index}=1\right)}^2}{2\left(x^2+2\right)}-\frac{\ln \left(\frac{3{\mathrm{RootOf}\left({\mathrm{\_Z}}^3-x^2-1\,\mathrm{index}=1\right)}^2+x^2+3\mathrm{RootOf}\left({\mathrm{\_Z}}^3-x^2-1\,\mathrm{index}=1\right)+2}{x^2+2}\right)}{2}$

$\mathrm{evala}\left(\mathrm{diff}\left(g\,x\right)-f\right)$

$0$

$f≔\mathrm{unapply}\left(x+y\,x\right)$

$f≔x\mapsto x+y$

$\mathrm{int}\left(f\,4..5\right)$

$\frac{9}{2}+y$

$\mathrm{int}\left(\frac{1}{x}\,x=a..2\right)$

Warning, unable to determine if 0 is between a and 2; try to use assumptions or use the AllSolutions option

${\int }_a^2\frac{1}{x}\ⅆx$

$r≔\mathrm{int}\left(\frac{1}{x}\,x=a..2\,'\mathrm{AllSolutions}'\right)$

$r≔\left\{\begin{array}{cc}\mathrm{undefined}& a<0\\ \mathrm{\infty }& a=0\\ -\ln \left(a\right)+\ln \left(2\right)& 0<a\end{array}\right.$

$r\mathbf{assuming}0<a$

$-\ln \left(a\right)+\ln \left(2\right)$

$r\mathbf{assuming}a<0$

$\mathrm{undefined}$

$\mathrm{int}\left(\mathrm{abs}\left(\sin \left(x\right)\right)\&comma;x=0..2\mathrm{\pi }m\right)\mathbf{assuming}m::\mathrm{integer}$

${\int }_0^{2\mathrm{\pi }m}\left|\sin \left(x\right)\right|\&DifferentialD;x$

$\mathrm{int}\left(\mathrm{abs}\left(\sin \left(x\right)\right)\&comma;x=0..2\mathrm{\pi }m\&comma;'\mathrm{AllSolutions}'\right)\mathbf{assuming}m::\mathrm{integer}$

$4m$

The int command was updated in Maple 2016; see Advanced Math Updates for Maple 2016.

The int command was updated in Maple 2021.

The method option was updated in Maple 2023.