int - Maple Programming Help

int

definite and indefinite integration

Calling Sequence

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

 > $f≔7{x}^{3}+3{x}^{2}+5x:$
 > $∫f\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 $\frac{{7}}{{4}}{}{{x}}^{{4}}{+}{{x}}^{{3}}{+}\frac{{5}}{{2}}{}{{x}}^{{2}}$ (1)
 > $∫\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${-}{\mathrm{cos}}{}\left({x}\right)$ (2)
 > $∫\frac{x}{{x}^{3}-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${-}\frac{{1}}{{6}}{}{\mathrm{ln}}{}\left({{x}}^{{2}}{+}{x}{+}{1}\right){+}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{{1}}{{3}}{}\left({2}{}{x}{+}{1}\right){}\sqrt{{3}}\right){+}\frac{{1}}{{3}}{}{\mathrm{ln}}{}\left({x}{-}{1}\right)$ (3)
 > $∫{ⅇ}^{-{x}^{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 $\frac{{1}}{{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{erf}}{}\left({x}\right)$ (4)

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

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

Compute definite integrals.

 > ${∫}_{0}^{\mathrm{π}}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${2}$ (6)
 > ${∫}_{0}^{\mathrm{∞}}{ⅇ}^{-{x}^{2}}\mathrm{ln}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${-}\frac{{1}}{{4}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{γ}}{-}\frac{{1}}{{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{ln}}{}\left({2}\right)$ (7)
 > ${∫}_{0}^{\mathrm{∞}}{ⅇ}^{-{x}^{2}}{\mathrm{ln}\left(x\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 $\frac{{1}}{{16}}{}{{\mathrm{π}}}^{{5}{/}{2}}{+}\frac{{1}}{{8}}{}{{\mathrm{γ}}}^{{2}}{}\sqrt{{\mathrm{π}}}{+}\frac{{1}}{{2}}{}{\mathrm{γ}}{}{\mathrm{ln}}{}\left({2}\right){}\sqrt{{\mathrm{π}}}{+}\frac{{1}}{{2}}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{}\sqrt{{\mathrm{π}}}$ (8)

An Elliptic integral

 > ${∫}_{2}^{3}\frac{1}{\sqrt{2{t}^{4}-3{t}^{2}-2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$
 $\frac{{1}}{{5}}{}\sqrt{{5}}{}{\mathrm{EllipticF}}{}\left(\frac{{1}}{{3}}{}\sqrt{{7}}{,}\frac{{1}}{{5}}{}\sqrt{{5}}\right){-}\frac{{1}}{{5}}{}\sqrt{{5}}{}{\mathrm{EllipticF}}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{,}\frac{{1}}{{5}}{}\sqrt{{5}}\right)$ (9)

A double integral

 > $\mathrm{int}\left(x{y}^{2},\left[x,y\right]\right)$
 $\frac{{1}}{{6}}{}{{x}}^{{2}}{}{{y}}^{{3}}$ (10)
 > ${∫}_{-2}^{2}{∫}_{0}^{y}x{y}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆxⅆy$
 $\frac{{32}}{{5}}$ (11)

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

 > ${∫}_{-2.0}^{2}{∫}_{0.0}^{y}x{y}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆxⅆy$
 ${6.400000000}$ (12)

An integral with decimal limits using numerical methods:

 > ${∫}_{0.75}^{1.25}\frac{x}{{x}^{3}+1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$
 ${0.2459707569}$ (13)

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

 > $\mathrm{int}\left(\frac{x}{{x}^{3}+1},x=0.75..1.25,\mathrm{numeric}=\mathrm{false}\right)$
 ${-}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{{1}}{{6}}{}\sqrt{{3}}\right){-}\frac{{1}}{{6}}{}{\mathrm{ln}}{}\left({13}\right){+}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({7}\right){+}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{arctan}}{}\left(\frac{{1}}{{2}}{}\sqrt{{3}}\right){-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({3}\right)$ (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:
 • Integration over a complex interval
 • Numerical integration
 • Handling discontinuities
 • Series expansions
 • Inert form of the int command, Int
 see the int/details help page.

Compatibility

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