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keywords=anti-derivative, area, areas, calculus, differential equations, Dwight s tables, Gradshteyn-Ryzhik, int, Int, integrate, Integrate, integration, integration over regions, intparts, Marichev-Adamchik methods, measures, Mellin transform methods, mensuration, multiple integrals, polyint, region integrals, Risch algorithm, Slater s Theorem, surface integrals, volume integrals, volumes;
description=Integrate[f, x] gives the indefinite integral \[Integral]f d x. Integrate[f, x, xmin, xmax ] gives the definite integral \[Integral]_xmin^xmax\ f\ d x. Integrate[f, x, xmin, xmax , y, ymin, ymax , ...] gives the multiple integral \[Integral]_xmin^xmaxd x \[Integral]_ymin^ymaxd y\ \ ...\ f. Integrate[f, x, y, ... \[Element] reg] integrates over the geometric region reg.;

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led neoncolors dashedlines similar integrals can lead to functions of different kinds wolfram language code integrate sqrt x sqrt 1 x x wolfram language code integrate sqrt x sqrt 1 x sqrt 2 x x many integrals can be done only in terms of special functions such as erf wolfram language code integrate exp x 2 x generalizations of log such as polylog and logintegral wolfram language code integrate log 1 x 2 x x wolfram language code integrate log log x x hypergeometric functions such as hypergeometric2f1 wolfram language code integrate tan x n x create a nicely formatted table of special function integrals wolfram language code flist besselj 0 x erf x elliptice x sqrt x polylog 2 x airyai x sinhintegral 2 x 1 wolfram language code grid prepend transpose flist integrate flist x f x f x ⅆx iconizedobject grid options traditionalform the variable of integration need not be a single symbol wolfram language code integrate p x log p x p x definite integrals 13 integrate a polynomial wolfram language code integrate x 4 x 2 1 x 1 3 visualize the area under the curve wolfram language code plot x 4 x 2 1 x 1 3 filling axis integrate a symbolic polynomial wolfram language code integrate a x 2 b x c x 2 2 integrate over a symbolic range wolfram language code integrate x n n x x 0 a assumptions n 1 n ℤ rational functions wolfram language code integrate 1 x 4 x 2 1 x 0 infinity wolfram language code plot 1 x 4 x 2 1 x 0 5 filling axis plotrange all epilog rgbcolor 0 368417 0 506779 0 709798 arrow 0 0 5 0 wolfram language code integrate 4x 2 7x 12 x 2 x 3 x 1 2 wolfram language code plot 4 x 2 7 x 12 x 2 x 3 x 1 2 filling 0 plotrange all algebraic functions wolfram language code integrate 1 x 3 cuberoot x x 1 1 wolfram language code plot 1 x 3 cuberoot x x 1 1 filling axis fillingstyle rgbcolor 0 95 0 627 0 1425 0 2 rgbcolor 0 24 0 6 0 8 0 2 wolfram language code integrate 1 2 x 2 sqrt 4 3x 2 x wolfram language code plot 1 2 x 2 sqrt 4 3 x 2 x 7 7 filling 0 axesorigin 0 0 epilog rgbcolor 0 368417 0 506779 0 709798 arrowheads medium medium arrow 7 0 7 0 trigonometric functions wolfram language code integrate cos 2x 4 x 0 π wolfram language code plot cos 2x 4 x 0 π 2 filling axis wolfram language code integrate x 2sin 2x x 0 2π wolfram language code plot x 2sin 2x x 0 2π filling axis fillingstyle rgbcolor 0 95 0 627 0 1425 0 2 rgbcolor 0 24 0 6 0 8 0 2 exponential and logarithmic functions wolfram language code integrate x 5log x x 1 2 wolfram language code plot x 5log x x 1 2 filling axis wolfram language code integrate exp x exp x x 0 wolfram language code plot exp x exp x x 0 4 plotrange full filling axis epilog rgbcolor 0 368417 0 506779 0 709798 arrowheads medium arrow 0 0 4 0 hyperbolic trigonometric functions wolfram language code integrate x cosh x x 0 a wolfram language code integrate x 2 sech x 2 x integrate a function with a vertical asymptote wolfram language code integrate x sqrt 1 x x 0 1 wolfram language code plot x sqrt 1 x x 0 1 filling axis this can be viewed as a limit of the result of integration on a smaller interval wolfram language code underscript a 1 subsuperscript 0 a x sqrt 1 x ⅆx compute the integral of a function with two vertical asymptotes wolfram language code area subsuperscript 1 1 x 2 sqrt 1 x 4 ⅆx wolfram language code n wolfram language code plot x 2 sqrt 1 x 4 x 1 1 filling 0 this can be viewed as a multivariate limit of the result of integration on a smaller interval wolfram language code area limit integrate x 2 sqrt 1 x 4 x a b assumptions 1 a b 1 a b 1 1 direction fromabove frombelow integrals over infinite intervals can be viewed as limits of integrals over finite domains wolfram language code integrate x e x x 1 the preceding is the limit as of the integral from to wolfram language code underscript a integrate x e x x 1 a an integral over reals wolfram language code subsuperscript e x 2ⅆx it is the bivariate limit of a finite integral wolfram language code underscript a b subsuperscript a b e x 2ⅆx when there are parameters conditions that ensure convergence may be reported wolfram language code integrate x n x 0 1 wolfram language code integrate e a x x 0 integrals of elementary functions may produce special function answers wolfram language code integrate sin t t t a b wolfram language code integrate cos sin x 2 x 0 2π wolfram language code n wolfram language code plot cos sin x 2 x 0 2π filling 0 plotrange 0 1 create a formatted table of definite integrals over the positive reals of special functions wolfram language code flist besselj 2 x x besselk 0 x 2 airyai x 2 exp x sinc x sin x erfc x wolfram language code grid prepend transpose flist subsuperscript 0 flistⅆx f x subsuperscript 0 f x ⅆx iconizedobject grid options traditionalform integral along a complex line wolfram language code integrate sqrt x x i 3 i integrate along a piecewise linear contour in the complex plane wolfram language code integrate 1 z 1 2 z 1 e i π 3 e 2 i π 3 1 e 2 i π 3 e i π 3 1 fullsimplify integrate the same function along a circular contour wolfram language code integrate i exp i ω exp i ω 1 2 ω 0 2π plot the function and paths of integration wolfram language code complexplot 1 z 1 2 z 1 2 1 2i 1 2 1 2i epilog thick arrowheads medium white arrow 1 05 1 0 circle dashed black arrow reim exp 2π i 6 range 0 6 integrals of piecewise and generalized functions 12 compute the indefinite integral of a piecewise function wolfram language code f x_ piecewise x 2 x 0 x 2 true wolfram language code f x ⅆx in this case the derivative of the integral equals the original function wolfram language code simplify d x f x integrate a discontinuous piecewise function wolfram language code f x_ piecewise x 1 x 0 x 0 x 1 x 2 true wolfram language code g x_ f x ⅆx except at the point of discontinuity the derivative of g equals f wolfram language code resolve forall x x 0 x ℝ derivative 1 g x f x visualize the function and its antiderivative wolfram language code plot f x g x x 2 2 plottheme detailed dashedlines integrate functions that are piecewise defined wolfram language code unitint x_ integrate x unitstep x x wolfram language code plot x unitstep x unitint x x 2 2 plottheme detailed dashedlines wolfram language code clipint x_ integrate clip x 2 x wolfram language code plot clip x 2 clipint x x 2 2 plottheme detailed dashedlines integrate a piecewise function with infinitely many cases wolfram language code maxint x_ integrate max sin x cos x x everywhere it is defined the derivative of maxint equals the original function wolfram language code simplify d maxint x x max sin x cos x however maxint itself is discontinuous wolfram language code plot max sin x cos x maxint x x 0 4pi plottheme detailed dashedlines compute a definite integral of a piecewise function wolfram language code f x_ piecewise 1 x 2 x 3 2 x 3 0 wolfram language code integrate f x x 0 3 compute the integral with a variable endpoint wolfram language code int x_ integrate f t t 0 x assumptions x 0 visualize the function and its integral wolfram language code plot f x int x x 0 4 plottheme detailed dashedlines compute definite integrals of piecewise functions such as floor wolfram language code integrate floor x 2 x 0 3 wolfram language code plot floor x 2 x 0 3 filling axis primepi wolfram language code integrate primepi n n 0 100 wolfram language code plot primepi x x 0 100 filling axis a composition of piecewise functions wolfram language code integrate ceiling x 2 trianglewave x x 2 2 compute the definite integral with a variable upper limit wolfram language code f x_ x 2 unitstep x 1 wolfram language code int a_ integrate f x x 3 a wolfram language code plot f a int a a 2 2 plottheme detailed dashedlines a function with an infinite number of cases wolfram language code f x_ max sin x cos x wolfram language code f x piecewiseexpand integrate over a finite number of cases using assumptions wolfram language code maxint a_ integrate f x x 0 a assumptions 0 a 2pi the integral is a continuous function of the upper limit over the domain of integration wolfram language code plot f a maxint a a 0 2pi plottheme detailed dashedlines integrate generalized functions wolfram language code integrate diracdelta x x wolfram language code integrate sin x 2 heavisidetheta x π heavisidetheta π x x wolfram language code integrate diracdelta cos x 2 1 x 2 2 indefinite integrals of generalized functions return generalized functions wolfram language code integrate diracdelta x x a nested integral wolfram language code integrate diracdelta x x x integrate generalized functions over subsets of the reals wolfram language code integrate f x diracdelta x a x 1 1 assumptions a reals wolfram language code integrate cos x a heavisidelambda x x 0 pi assumptions a reals integrate an interpolating function wolfram language code f interpolation 10 5 10 10 20 wolfram language code g head integrate f x x test that g is a correct antiderivative at x 3 5 wolfram language code g 3 5 f 3 5 visualize the function and its antiderivative wolfram language code plot f x g x x 1 5 plottheme detailed dashedlines nested integrals 12 compute a second antiderivative of a function wolfram language code integrate a x 2 b x c x x expand wolfram language code d x 2 compute the third antiderivative wolfram language code 1 x 2 4 ⅆxⅆxⅆx wolfram language code fullsimplify d x 3 integrate a function with respect to two different variables wolfram language code integrate x 2sin y y x the mixed partial derivative gives the original function wolfram language code d x y generate a constant of integration for a single integral wolfram language code integrate x x generatedparameters c generate constants for a nested integral with respect to the same variable wolfram language code integrate x x x generatedparameters c this is the most general second antiderivative of the integrand wolfram language code d x x generate two functions of integration for a nested integral with respect to two variables wolfram language code integrate x x y generatedparameters c this is the most general mixed antiderivative of the integrand wolfram language code d x y integrate over the rectangle from to wolfram language code integrate exp x cos y x 0 a y 0 b equivalently wolfram language code subsuperscript 0 a subsuperscript 0 b exp x cos y ⅆyⅆx integrate in the opposite order wolfram language code subsuperscript 0 b subsuperscript 0 a exp x cos y ⅆxⅆy combine indefinite and definite integration wolfram language code integrate x y y 0 1 x compute a rational double integral over a rectangular region wolfram language code integrate x 2y x y x 0 3 y 1 2 this gives the volume of the shaded region wolfram language code plot3d x 2 y x y x 0 3 y 1 2 filling axis fillingstyle opacity 0 6 mesh none compute a trigonometric double integral over a rectangular region wolfram language code integrate y sin x y x 1 4 y 0 π there is as much positive volume dark gray as negative light blue wolfram language code plot3d y sin x y 0 x 1 4 y 0 π iconizedobject plot3d options compute a polynomial double integral over the area between two curves wolfram language code subsuperscript 2 2 subsuperscript 2 x 2 4 x 2 8 3 y 3 3 x y 3 x 2 y 3 x 3 y 3 ⅆyⅆx visualize the domain of integration and the volume corresponding to the integral wolfram language code plot 2x 2 4 x 2 x 2 2 filling 1 2 plotlegends expressions wolfram language code plot3d 8 3 y 3 3 x y 3 x 2 y 3 x 3 y 3 x y implicitregion 2 x 2 2 x 2 y 4 x 2 x y fillingstyle opacity 5 filling axis plotrange all viewpoint 3 3 1 1 3 mesh none compute a triple integral over a rectangular prism wolfram language code integrate cos z exp x y x 0 1 y 1 2 z 0 3 visualize the region of integration wolfram language code regionplot3d 0 x 1 1 y 2 0 z 3 x 0 5 1 5 y 1 5 2 5 z 0 5 3 5 compute an algebraic triple integral over a cone shaped 3d region wolfram language code integrate sqrt x 2 z 2 y x 2 2 y x 2 4 z sqrt y x 2 sqrt y x 2 fullsimplify visualize the region of integration wolfram language code regionplot3d y x 2 z 2 0 y 4 x 3 3 y 0 5 4 5 z 3 3 integrate a multivariate function over a five dimensional cube wolfram language code integrate a b 2 c d e a 0 1 b 1 2 c 2 3 d 3 4 e 4 5 wolfram language code n integrate over the unit ball in 4 dimensions wolfram language code f r_ θ_ sin θ r look up the coordinate ranges for hyperspherical coordinates in coordinatechartdata wolfram language code coordinatechartdata hyperspherical coordinaterangeassumptions r θ φ ψ also look up the volume factor wolfram language code coordinatechartdata hyperspherical volumefactor r θ φ ψ do the integral wolfram language code subsuperscript π π subsuperscript 0 π subsuperscript 0 π subsuperscript 0 r f r θ r 3 sin θ 2 sin φ ⅆrⅆθⅆφⅆψ region integrals 11 integrate a constant over a unit disk wolfram language code integrate 1 x y disk enter the integral in typeset form wolfram language code underscript x y disk 1 equivalently integrate over a rectangular region and restrict to a disk using boole wolfram language code integrate boole x 2 y 2 1 x 1 1 y 1 1 an integral over a unit disk wolfram language code subscript x y disk cos 3x sin 5y x 2 the same integral expressed using boole wolfram language code subsuperscript 1 1 subsuperscript 1 1 cos 3x sin 5y x 2 boole x 2 y 2 1 ⅆyⅆx the same integral reduced to an iterated integral with bounds depending on the previous variables wolfram language code subsuperscript 1 1 subsuperscript sqrt 1 x 2 sqrt 1 x 2 cos 3x sin 5y x 2 ⅆyⅆx plot the integrand over the integration region wolfram language code plot3d cos 3x 2sin 5y 1 x y disk filling 0 mesh none fillingstyle automatic express a normal definite integral using region notation wolfram language code integrate sin b x x interval 0 pi compare with the list notation wolfram language code integrate sin b x x 0 pi with generateconditions true assumptions are generated so that the region is non degenerate wolfram language code integrate x 2y x y rectangle 0 0 a b generateconditions true compare with an iterated integral wolfram language code integrate x 2y x 0 a y 0 b generateconditions true integrate over the unit circle wolfram language code subscript x y circle x 2 y y 2 2 express the same integral as a one dimensional integral using polar coordinates wolfram language code subsuperscript pi pi cos θ 2sin θ sin θ 2 2 ⅆθ integrate over a sphere of radius wolfram language code underscript x y z sphere x0 y0 z0 r 1 x x0 2 y y0 2 z z0 2 integrate over a finite set of points wolfram language code integrate 1 x y z point 1 2 3 4 5 6 7 8 9 regions can be given as logical combinations of inequalities wolfram language code int1 integrate boole 1 x 2 y 2 4 x y 1 x 0 y 0 x y define the region as an implicitregion and integrate directly over the region wolfram language code reg implicitregion 1 x 2 y 2 4 x y 1 x 0 y 0 x y wolfram language code int2 integrate 1 x y reg the integrals are equivalent wolfram language code fullsimplify int1 int2 visualize the domain of integration ...