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 $\int f\ⅆx$.

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.

$f≔7x^3\+3x^2\+5x\:$

$\∫f\ⅆx$

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

$\∫\sin \left(x\right)\ⅆx$

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

$\∫\frac{x}{x^3-1}\ⅆx$

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

$\∫{\ⅇ}^{-x^2}\ⅆx$

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

$\∫{\ⅇ}^{-x^2}\ln \left(x\right)\ⅆx$

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

${\∫}_0^{\mathrm{Pi}}\sin \left(x\right)\ⅆx$

$2$

${\∫}_0^{\mathrm{\∞}}{\ⅇ}^{-x^2}\ln \left(x\right)\ⅆx$

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

${\∫}_0^{\mathrm{\∞}}{\ⅇ}^{-x^2}{\ln \left(x\right)}^2\ⅆx$

$\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}$

${\∫}_2^3\frac{1}{\sqrt{2t^4-3t^2-2}}\ⅆt$

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

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

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

${\∫}_{-2}^2{\∫}_0^{y}x{y}^2\ⅆx\ⅆy$

$\frac{32}{5}$

${\∫}_{-2.0}^2{\∫}_{0.0}^{y}x{y}^2\ⅆx\ⅆy$

$6.400000000$

${\∫}_{0.75}^{1.25}\frac{x}{x^3\+1}\ⅆx$

$0.2459707569$

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

$-\frac{\sqrt{3}\arctan \left(\frac{\sqrt{3}}{6}\right)}{3}-\frac{\ln \left(13\right)}{6}+\frac{\ln \left(7\right)}{2}+\frac{\sqrt{3}\arctan \left(\frac{\sqrt{3}}{2}\right)}{3}-\frac{\ln \left(3\right)}{2}$

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

$0.2459707569$

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.