Integral Calculus - Maple Help

Home : Support : Online Help : Mathematics : Calculus : Integration : Integral Calculus

int

definite and indefinite integration

Calling Sequence

 int(expression,x, options) $\int \mathrm{expression}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$ int(expression,x=a..b, options) ${\int }_{a}^{b}\mathrm{expression}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$ int(expression, [x, y, ...], options) $\int \int \mathrm{expression}ⅆxⅆy$ int(expression, [x = a..b, y = c..d, ...], options) ${\int }_{c}^{d}{\int }_{a}^{b}\mathrm{expression}ⅆxⅆy$

Parameters

 expression - algebraic expression; integrand x, y - names; variables of integration a, b, c, d - endpoints of interval on which integral is taken options - (optional) various options to control the type of integration performed. For example, numeric=true will perform numeric instead of symbolic integration. See int/details for more options.

Description

 • The int(expression, x) calling sequence computes an indefinite integral of the expression with respect to the variable x. Note: No constant of integration appears in the result.
 • The int(expression, x = a..b) calling sequence computes the definite integral of the expression with respect to the variable x on the interval from a to b.
 • The int(expression, [ranges or variables]) calling sequence computes the iterated definite integral of the expression with respect to the variables or ranges in the list in the order they appear in the list. Note: The notation int(expression, [x = a..b, y = c..d]) is equivalent to int(int(expression, x = a..b), y = c..d) except that the single call to int accounts for the range of the outer variables (via assumptions) when computing the integration with respect to the inner variables.
 • You can enter the command int using either the 1-D or 2-D calling sequence.  For example, int(f,x) is equivalent to $\int f\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$.
 • If any of the integration limits of a definite integral are floating-point numbers (e.g. 0.0, 1e5 or an expression that evaluates to a float, such as exp(-0.1)), then int computes the integral using numerical methods if possible (see evalf/int). Symbolic integration will be used if the limits are not floating-point numbers unless the numeric=true option is given.
 • If Maple cannot find a closed form expression for the integral (or the floating-point value for definite integrals with float limits), the function call is returned.
 • Note: For information on the inert function, Int, see int/details.

Examples

No constant of integration appears in the result for indefinite integrals.

 > $f≔7{x}^{3}+3{x}^{2}+5x:$
 > $\mathrm{int}\left(f,x\right)$
 $\frac{{7}}{{4}}{}{{x}}^{{4}}{+}{{x}}^{{3}}{+}\frac{{5}}{{2}}{}{{x}}^{{2}}$ (1)
 > $\mathrm{int}\left(\mathrm{sin}\left(x\right),x\right)$
 ${-}{\mathrm{cos}}{}\left({x}\right)$ (2)
 > $\mathrm{int}\left(\frac{x}{{x}^{3}-1},x\right)$
 ${-}\frac{{\mathrm{ln}}{}\left({{x}}^{{2}}{+}{x}{+}{1}\right)}{{6}}{+}\frac{\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{\left({2}{}{x}{+}{1}\right){}\sqrt{{3}}}{{3}}\right)}{{3}}{+}\frac{{\mathrm{ln}}{}\left({x}{-}{1}\right)}{{3}}$ (3)
 > $\mathrm{int}\left(\mathrm{exp}\left(-{x}^{2}\right),x\right)$
 $\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{erf}}{}\left({x}\right)}{{2}}$ (4)

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

 > $\mathrm{int}\left(\mathrm{exp}\left(-{x}^{2}\right)\mathrm{ln}\left(x\right),x\right)$
 ${\int }{{ⅇ}}^{{-}{{x}}^{{2}}}{}{\mathrm{ln}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (5)

Compute definite integrals.

 > $\mathrm{int}\left(\mathrm{sin}\left(x\right),x=0..\mathrm{\pi }\right)$
 ${2}$ (6)
 > $\mathrm{int}\left(\mathrm{exp}\left(-{x}^{2}\right)\mathrm{ln}\left(x\right),x=0..\mathrm{\infty }\right)$
 ${-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\gamma }}}{{4}}{-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{ln}}{}\left({2}\right)}{{2}}$ (7)
 > $\mathrm{int}\left(\mathrm{exp}\left(-{x}^{2}\right){\mathrm{ln}\left(x\right)}^{2},x=0..\mathrm{\infty }\right)$
 $\frac{{{\mathrm{\pi }}}^{{5}}{{2}}}}{{16}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}{}\sqrt{{\mathrm{\pi }}}}{{8}}{+}\frac{{\mathrm{\gamma }}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{ln}}{}\left({2}\right)}{{2}}{+}\frac{\sqrt{{\mathrm{\pi }}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}}{{2}}$ (8)

An Elliptic integral

 > $\mathrm{int}\left(\frac{1}{\mathrm{sqrt}\left(2{t}^{4}-3{t}^{2}-2\right)},t=2..3\right)$
 $\frac{\sqrt{{5}}{}{\mathrm{EllipticF}}{}\left(\frac{\sqrt{{7}}}{{3}}{,}\frac{\sqrt{{5}}}{{5}}\right)}{{5}}{-}\frac{\sqrt{{5}}{}{\mathrm{EllipticF}}{}\left(\frac{\sqrt{{2}}}{{2}}{,}\frac{\sqrt{{5}}}{{5}}\right)}{{5}}$ (9)

A double integral

 > $\mathrm{int}\left(x{y}^{2},\left[x,y\right]\right)$
 $\frac{{{x}}^{{2}}{}{{y}}^{{3}}}{{6}}$ (10)
 > $\mathrm{int}\left(x{y}^{2},\left[x=0..y,y=-2..2\right]\right)$
 $\frac{{32}}{{5}}$ (11)

If either of the integration limits are floating-point numbers, then int computes the integral using numerical methods.

 > $\mathrm{int}\left(x{y}^{2},\left[x=0...y,y=-2.0..2\right]\right)$
 ${6.400000000}$ (12)

An integral with decimal limits using numerical methods:

 > $\mathrm{int}\left(\frac{x}{{x}^{3}+1},x=0.75..1.25\right)$
 ${0.2459707569}$ (13)

To apply symbolic integration methods instead, use numeric=false:

 > $\mathrm{int}\left(\frac{x}{{x}^{3}+1},x=0.75..1.25,\mathrm{numeric}=\mathrm{false}\right)$
 ${-}\frac{\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{\sqrt{{3}}}{{6}}\right)}{{3}}{-}\frac{{\mathrm{ln}}{}\left({13}\right)}{{6}}{+}\frac{{\mathrm{ln}}{}\left({7}\right)}{{2}}{+}\frac{\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{\sqrt{{3}}}{{2}}\right)}{{3}}{-}\frac{{\mathrm{ln}}{}\left({3}\right)}{{2}}$ (14)

The option numeric=true or simply numeric may also be used to compute a numerical integral even with exact limits:

 > $\mathrm{int}\left(\frac{x}{{x}^{3}+1},x=\frac{3}{4}..\frac{5}{4},\mathrm{numeric}\right)$
 ${0.2459707569}$ (15)

Details

 For detailed information including:
 • Numerical integration
 • Integration involving Units
 • Handling discontinuities
 • Series expansions
 • Integration over a complex interval
 • Inert form of the int command, Int
 see the int/details help page.