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Integrate

Integrate[f,x]

gives the indefinite integral .

Integrate[f,{x,xmin,xmax}]

gives the definite integral .

Integrate[f,{x,xmin,xmax},{y,ymin,ymax},…]

gives the multiple integral .

Integrate[f,{x,y,…}∈reg]

integrates over the geometric region reg.

Details and Options

Examples

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Basic Examples  (4)

Indefinite integral:

Compute a definite integral:

Visualize the area given by this integral:

Use int to enter ∫ and dd to enter :

Use to enter the lower limit, then for the upper limit:

Scope  (77)

Basic Uses  (13)

Compute an indefinite integral:

Verify the answer by differentiation:

Use intt to enter a template and to move between fields:

Include the constant of integration in an indefinite integral:

Compute a definite integral over a finite interval:

Infinite interval:

Doubly infinite interval:

Use dintt to enter a template and to move between fields:

Integrate a function with a symbolic parameter:

An integral that only converges for some values of parameters:

Specify alternate assumptions to use:

Multivariate integrals:

Multiple integral with x integration last:

In StandardForm, the differential y precedes x:

Visualize the function over the domain of integration:

Integrals over standard regions:

The character ∈ can be entered as el or ∈:

Enter a region specification in an underscript using :

Use rintt to enter a template and to move between fields:

Formal integrals:

Integrals of vector- and array-valued functions:

Invoke NIntegrate automatically if symbolic integration fails:

Indefinite Integrals  (10)

Some basic integrals:

Generate an answer with a constant of integration:

Integrals of trigonometric functions:

Verify the previous answer via differentiation:

Create a nicely formatted table of integrals:

Rational functions can always be integrated in closed form:

Sometimes they involve sums of Root objects:

Integrals of general elementary functions:

Integrate returns antiderivatives valid in the complex plane where applicable:

A common antiderivative found in integral tables for is log(TemplateBox[{{sec, (, x, )}}, RealAbs]):

This is a valid antiderivative for real values of :

On the real line, the two integrals have the same real part:

But the imaginary parts differ by on any interval where is negative:

Similar integrals can lead to functions of different kinds:

Many integrals can be done only in terms of special functions such as Erf:

Generalizations of Log such as PolyLog and LogIntegral:

Hypergeometric functions such as Hypergeometric2F1:

Create a nicely-formatted table of special function integrals:

The variable of integration need not be a single symbol:

Definite Integrals  (13)

Integrate a polynomial:

Integrate a symbolic polynomial:

Integrate over a symbolic range:

Rational functions:

Algebraic functions:

Trigonometric functions:

Exponential and logarithmic functions:

Hyperbolic trigonometric functions:

Integrate a function with a vertical asymptote:

This can be viewed as a limit of the result of integration on a smaller interval:

Compute the integral of a function with two vertical asymptotes:

This can be viewed as a multivariate limit of the result of integration on a smaller interval:

Integrals over infinite intervals can be viewed as limits of integrals over finite domains:

The preceding is the limit as of the integral from to :

An integral over reals:

It is the bivariate limit of a finite integral:

When there are parameters, conditions that ensure convergence may be reported:

Integrals of elementary functions may produce special function answers:

Create a formatted table of definite integrals over the positive reals of special functions:

Integral along a complex line:

Along a piecewise linear contour in the complex plane:

Along a circular contour in the complex plane:

Plot the function and paths of integration:

Integrals of Piecewise and Generalized Functions  (12)

Compute the indefinite integral of a Piecewise function:

In this case, the derivative of the integral equals the original function:

Integrate a discontinuous Piecewise function:

Except at the point of discontinuity, the derivative of g equals f:

Visualize the function and its antiderivative:

Integrate functions that are piecewise-defined:

Integrate a piecewise function with infinitely many cases:

Everywhere the derivative is defined, the derivative of maxInt equals the original function:

However, maxInt itself is discontinuous:

Compute a definite integral of a Piecewise function:

Compute the integral with a variable endpoint:

Visualize the function and its integral:

Compute definite integrals of piecewise functions such as Floor:

PrimePi:

A composition of piecewise functions:

Compute the definite integral with a variable upper limit:

A function with an infinite number of cases:

Integrate over a finite number of cases using Assumptions:

The integral is a continuous function of the upper limit over the domain of integration:

Integrate generalized functions:

Indefinite integrals of generalized functions return generalized functions:

A nested integral:

Integrate generalized functions over subsets of the reals:

Integrate an interpolating function:

Test that g is a correct antiderivative at x==3.5:

Visualize the antiderivative:

Nested Integrals  (11)

Compute a second antiderivative of a function:

Compute the third antiderivative:

Integrate a function with respect to two different variables:

The mixed partial derivative gives the original function:

Generate a constant of integration for a single integral:

Generate constants for a nested integral with respect to the same variable:

This is the most general second antiderivative of the integrand:

Generate two functions of integration for a nested integral with respect to two variables:

This is the most general mixed antiderivative of the integrand:

Integrate over the rectangle from to :

Equivalently:

Integrate in the opposite order:

Combine indefinite and definite integration:

Compute a rational double integral over a rectangular region:

This gives the volume of the shaded region:

Compute a trigonometric double integral over a rectangular region:

There is as much positive volume (dark gray) as negative (light blue):

Compute a polynomial double integral over the area between two curves:

Visualize the domain of integration and the volume corresponding to the integral:

Compute a triple integral over a rectangular prism:

Visualize the region of integration:

Integrate a multivariate function over a five-dimensional cube:

Integrate over the unit ball in 4 dimensions:

Look up the coordinate ranges for hyperspherical coordinates in CoordinateChartData:

Also look up the volume factor:

Do the integral:

Region Integrals  (11)

Integrate a constant over a unit disk:

Enter the integral in typeset form:

Equivalently, integrate over a rectangular region and restrict to a disk using Boole:

An integral over a unit disk:

The same integral expressed using Boole:

The same integral reduced to an iterated integral with bounds depending on the previous variables:

Plot the integrand over the integration region:

Express a normal definite integral using region notation:

Compare with the list notation:

With symbolic endpoints, assumptions are generated so that the region is non-degenerate:

Integrate over the unit circle:

Express the same integral as a one-dimensional integral using polar coordinates:

Integrate over a sphere of radius :

Integrate over a finite set of points:

Regions can be given as logical combinations of inequalities:

Define the region as an ImplicitRegion and integrate directly over the region:

The integrals are equivalent:

Visualize the domain of integration:

Integral over a three-dimensional region defined by inequalities:

Visualize 3D regions using RegionPlot3D:

Integrate over a solid cone:

Visualize the domain of integration:

Integrate a function with parameters, getting a piecewise result:

A region with infinitely many components:

Symbolic Features of Integrals  (7)

Integrals involving unknown functions are done when possible:

Differentiate with respect to an endpoint, yielding the fundamental theorem of calculus:

A generalization:

Symbolic integrals can be differentiated with respect to parameters:

Differentiate with respect to a parameter that appears in both integrand and endpoints:

Use the Inactive form of Integrate:

Differentiate:

Illustrate indefinite integral identities:

Verify the identities starting from the inactive forms:

Illustrate the basic commutation trick for differentiating under the integral sign:

Compute the LaplaceTransform of an integral:

Options  (11)

Assumptions  (3)

By default, conditions are generated on parameters that guarantee convergence:

With Assumptions, a result valid under the given assumptions is given:

Manually specify Assumptions to test values outside the automatically generated conditions:

This integral is also convergent for purely imaginary :

Specify assumptions to evaluate a piecewise indefinite integral:

GenerateConditions  (2)

By default, univariate definite integrals generate conditions on parameters that ensure convergence:

Generate a result without conditions:

Use GenerateConditions->False to speed up integration:

GeneratedParameters  (4)

By default a particular antiderivative is returned:

Specify a value of GeneratedParameters to obtain the general antiderivative:

One parameter is generated for each indefinite integral:

If the input expression already contains a generated parameter, the next available index will be used:

For nested integrals with multiple variables, the antiderivative contains arbitrary functions:

This is the most general antiderivative of the integrand:

The value of GeneratedParameters is applied to the index of each generated parameter:

The value can be a pure function:

A value of None disables generated parameters:

PrincipalValue  (2)

The ordinary Riemann definite integral is divergent:

The Cauchy principal value integral is finite:

The value is the limit of removing a symmetric region about the singularity:

The ordinary Riemann definite integral is divergent:

Regularize the divergence at :

Applications  (67)

The Geometry of Integrals  (5)

The integral of a constant function is the signed area of the rectangle of height and width :

Visualize the two rectangles:

The integral of a piecewise-constant function is the sum of the signed areas of the rectangles defined by its plot:

Visualize the rectangles:

The integral of a general function is the signed area between its plot and the horizontal axis:

This can be related to the piecewise-constant case by considering rectangles defined by its plot:

For n5 on the interval [0,2], the rectangles are the following:

The area of these rectangles defines a Riemann sum that approximates the area under the curve:

Using DiscreteLimit to obtain the exact answer as gives the same answer as Integrate did:

Visualize the process for this function as well as three others:

The Fundamental Theorem of Calculus relates a function to its integral from a fixed lower limit to a variable upper limit:

Consider the definite integral of the this from from to :

The Fundamental Theorem of Calculus states that :

This can be seen from the limit definition of derivative:

Note that is an area consisting of a rectangle of height and width plus a small correction that vanishes as , as illustrated by the following table for :

Hence, the limit can be seen geometrically to equal , as illustrated in the following visualization:

Integrate a discrete set of data with Interpolation:

Area Between Curves  (7)

Compute the area under the curve of from to :

Find the area under the curve of from to :

Determine the total area enclosed between of and the -axis:

The total area is given by the integral of the absolute value:

Equivalently compute this as the sum of two integrals of the difference between the top and bottom:

Compute the area between and from to :

Find the area enclosed by and :

Since , will be above in the interval of interest and the area will equal:

Visualize the region of interest and the two functions:

Compute the area enclosed by and :

Find the area as the integral of the absolute value of the difference over the entire interval:

Visualize the two functions and the area between them:

Use the plot the split the integral into two equivalent integrals with no absolute value:

To compute the area enclosed by , , and , first find the points of intersection:

Visualize the three curves over an area containing the points:

From the plot, it is clear is above the line and below the other two curves:

Area can be found using two integrals, one for each "top function":

This can be reduced to a single integral using Min:

Compare with the answer returned by Area:

Regions of Revolution  (7)

Compute the volume enclosed when for is rotated about the -axis:

Visualize the solid:

Use cylindrical shells to find the volume enclosed when , , is rotated about the -axis:

Visualize the solid, adding the cap at :

Find the volume of the region formed by rotating the area between and about the -axis:

Find where the curves intersect:

Between these two values of , is above :

Visualize the volume:

Integrate cylindrical shells of height and circumference to find the volume:

Determine the volume the region above and below for , rotated about the -axis:

Find where the curves intersect, adding the constraint on the range of :

The relevant range of values is between these two points:

Visualize the volume:

Integrate washers of area to find the volume:

Compute the surface area when for is rotated about the -axis:

Apply the formula of the infinitesimal width of each strip:

Multiply the width by the circumference of each circle and integrate:

Find the area when for - is rotated about the -axis:

The infinitesimal width of each strip is given by the following:

Multiplying the width by the circumference and integrating yields the answer:

Determine the surface area when for is rotated about line :

The infinitesimal width of each strip is given by the following:

Since for the curve in question, each strip has radius and width :

Find the numerical approximation of this value:

Visualize the surface using modified cylindrical coordinates based on the line , :

Arc Length, Surface Area, and Volume  (8)

Compute the arc length of the plot from to :

Apply the formula for infinitesimal arc length:

Integrate to find the arc length:

Compare with the answer returned by ArcLength:

Compute the arc length of the plot from to :

Apply the formula for infinitesimal arc length:

Integrate to find the arc length:

Compare with the answer returned by ArcLength:

Length of a parametrically defined circle:

The relevant parameter range is to :

The infinitesimal arc length is constant:

Integrate to find the total arc length:

Compare with the answer returned by ArcLength:

Length of a 3D parametrically defined ellipse:

Visualize the ellipse:

The infinitesimal arc length is non-constant:

Integrate to find the total arc length:

Compare with the answer returned by ArcLength:

Find the surface area of the plot over the rectangle :

Apply the formula for infinitesimal surface area of a plot:

Integrate to find the arc length:

Compare with the answer returned by Area:

Find the area of the surface where :

This surface is a torus:

Apply the formula for infinitesimal surface area of a parametric surface:

Integrate to find the total surface area:

Compare with the answer returned by Area:

Find the volume of the following parametric region, where , :

This region is a solid torus:

Compute the Jacobian determinant:

Integrate to find the volume:

Compare with the answer returned by Volume:

Find the volume of the following parametric region, where , , and :

The region is an ellipsoid:

Compute the Jacobian determinant:

Integrate to find the volume:

Compare with the answer returned by Volume:

Line Integrals  (6)

Compute the line integral of over the origin-centered ellipse with semi-major axes and :

Parameterize the ellipse:

Perform the integral using the fact that ds=TemplateBox[{{{c, ^, {(, ', )}}, (, t, )}}, Norm]dt:

Compare the direct integral over the ellipse:

Calculate the closed line integral of over the following parametric curve:

The curve forms an infinity figure, traversed from red to purple as shown in the following plot:

Define the vector field :

Perform the calculation using the definition :

To calculate ∫x4dx+x yy over the triangle with vertices , , and , define the associated vector field:

Parametrize the triangle using a piecewise-linear parametrization:

The parametrization is oriented counter-clockwise:

Compute the line integral from the definition :

Calculate the work done by the force as a particle takes the following path from , , to , :

Define the force field as function from points to vectors:

The work done is the line integral :

Find a potential function for the following vector field:

This is possible because the vector field is conservative:

Define a family of straight-line curves that go from the origin at time to at time :

Let be the line integral of from the origin to the point :

Verify that is a potential function for using Grad

Use Green's Theorem to find the area of the area enclosed by the following curve:

The following vector-field has a two-dimensional Curl of :

Apply Green's theorem in the form to compute the area:

Surface and Volume Integrals  (7)

Use Green's Theorem to compute over the circle centered at the origin with radius 3:

Visualize the vector field and circle for the line integral:

The circulation of the vector field can be computed using Curl:

Integrate over the interior of the circle:

Perform the integral using region notation:

Compute the integral over the unit sphere of :

Parameterize the sphere:

Determine infinitesimal surface area:

Perform the integral :

Compare with a region integral:

Verify Stoke's theorem for for the upper unit hemisphere:

Parameterize the surface using standard spherical coordinates:

Visualize the surface and the vector field:

The boundary of the surface is the unit circle in the -plane:

Compute the curl of the vector field:

Compute the oriented surface area element on the hemisphere:

Stoke's theorem, , states that line integral of on boundary equals the flux integral of its curl through the surface:

Use the divergence theorem to compute the flux of through the surface bounded above by , below by , and on the side by and :

The divergence theorem, , relates the flux to the volume integral of the divergence:

Use Gauss's Theorem to find the volume enclosed by the following parametric surface:

The oriented area element on the surface is given by the following:

The following vector-field has a divergence equal :

Apply Gauss's Theorem in the form to compute the volume:

Given a mass density , find the mass of region given by the following:

The ranges of the parameters are and , producing a filled torus:

Enter the mass density function:

Compute the Jacobian determinate:

Integrate to find the total mass:

Derive a formula for the integral of over an -dimensional unit ball:

Verify the formula:

Average Values and Centroids  (6)

Compute the average value of between and :

Visualize the function and its average value:

Find the mean of over the parallelogram based at the origin with sides and :

As , the mean is given by the following ratio of integrals:

Express the integrals using region notation:

Visualize the function and its mean value:

To compute the centroid of the region under the curve of from to , first find the area:

The centroid equals the average value of the coordinates:

Compare with the answer given by RegionCentroid:

Determine the centroid of the region between the curves and from to :

Compare with the answer returned by RegionCentroid:

Visualize the region and its centroid:

Derive general formulas for the centroid of the region under the curve from to using the fact that the integral gives the area under the curve:

The centroid is the mean value of over the region from to and from to :

The centroid is similarly the mean value of :

Find the center of mass of the origin-centered hemisphere of radius with :

First compute the volume of the region:

The center of mass is the average value of the position vector:

Visualize the center of mass:

Probability, Expectation, and Standard Deviation  (7)

Compute the probability that when follows a standard normal distribution:

Compare with the value returned by Probability:

Computing the probability that for an exponential distribution with mean :

Computing the probability that :

The corresponding probabilistic statements:

Compute the probability that a value is within two standard deviations of the mean in a normal distribution:

Compare with the answer returned by Probability:

The value is approximate :

This can be interpreted as saying that about of the entire area under the curve lies between and in the following plot:

Compute the expectation of sqrt(TemplateBox[{x}, Abs]) when follows a standard Cauchy distribution:

Compare with the answer returned by Expectation:

Mean and variance of the normal distribution:

Compare with the built in functions Mean and Variance:

Show that the standard deviation of an exponential distribution with mean μ is also μ:

Compare with the answers returned by Mean and StandardDeviation:

Compute the cumulative distribution function (CDF) from the probability density function (PDF):

The CDF gives the area under the PDF curve from to :

Integral Transforms  (7)

Compute a Fourier transform:

Compare with FourierTransform:

Find a Laplace transform:

Compare with LaplaceTransform:

Define the Hartley transform:

Since the function is even, the Hartley transform is equivalent to FourierCosTransform:

Find the Fourier coefficients of a function on [0,1]:

Define the partial sums of the transform:

Visualize the partial sums, which exhibit the Gibbs phenomenon due to the a periodicity of :

Compute a Mellin transform:

Compare with MellinTransform:

Find a Hankel transform:

Compare with HankelTransform:

Compute a quadratic fractional Fourier transform in closed form:

Visualize the real and imaginary parts of the transform for different values of α:

Real and Complex Analysis  (4)

Define the standard L^p(TemplateBox[{}, Reals]) norm of a univariate function:

Also define a formatting for this function:

Compute the norms as a function of for three different functions:

The norm is always eventually an increasing function of , but it may be initially decreasing:

The Fourier transform is an isomorphism (the norm of the function and its transform are equal):

It is not, however, an isomorphism for any other value, for example for :

Define the weighted inner product for , with weight for functions defined on :

Orthogonality of Legendre polynomials TemplateBox[{n, x}, LegendreP] on with weight function 1:

Orthogonality of Chebyshev polynomials on with weight function :

Orthogonality of Hermite polynomials on with weight function :

Define an inner product on functions using Integrate:

Construct an orthonormal basis using using Orthogonalize:

This inner product produces the Gegenbauer polynomials:

Compute the residue of at as an integral over a contour enclosing :

Compare with the answers returned by Residue:

Integral Representation of Special Functions  (3)

Represent HermiteH in terms of Integrate:

Visualize the first five Hermite polynomials:

Express Gamma in terms of a logarithmic integral:

Visualize the function:

Represent Zeta in terms of Integrate:

Properties & Relations  (14)

Integration is a linear operator:

Indefinite integration is the inverse of differentiation:

Definite integration can be defined in terms of DiscreteLimit and Sum:

Evaluate integrals numerically using N:

This effectively calls NIntegrate:

Derivative with a negative integer order does integrals:

ArcLength is the integral of 1 over a one-dimensional region:

Area is the integral of 1 over a two-dimensional region:

Volume is the integral of 1 over a three-dimensional region:

RegionMeasure for a region is given by the integral :

RegionCentroid is equivalent to Integrate[p,p∈ℛ]/m with m=RegionMeasure[ℛ]:

Solve a simple differential equation:

DSolveValue returns a solution with the constant of integration:

DSolve returns a substitution rule for the solution:

Integrate computes the integral in closed form:

AsymptoticIntegrate gives series approximating the exact result:

FourierTransform is defined in terms of an integral:

LaplaceTransform is defined in terms of an integral:

Possible Issues  (12)

Indefinite Integrals  (6)

Many simple integrals cannot be evaluated in terms of standard mathematical functions:

The indefinite integral of a continuous function can be discontinuous:

Using a definite integral with a variable upper limit can smooth the discontinuity:

The derivative of an integral may not come out in the same form as the original function:

Simplify and related constructs can often show equivalence:

Different forms of the same integrand can give integrals that differ by constants of integration:

Parameters like are assumed to be generic inside indefinite integrals:

Use definite integration with a variable upper limit to generate conditions:

When part of a sum cannot be integrated explicitly, the whole sum will stay unintegrated:

Compare with:

Definite Integrals  (6)

Substituting limits into an indefinite integral may not give the correct result for a definite integral:

The presence of a discontinuity in the expression for the indefinite integral leads to the anomaly:

Specifying integer assumptions may not give a simpler result:

Use Simplify and related functions to obtain the expected result:

A definite integral may have a closed form only over an infinite interval:

Integrals over regions do not test whether an integrand is absolutely integrable:

Answers may then depend on how the region was decomposed for integration:

Integrals over zero-dimensional regions use the counting measure:

To use the measure of the ambient space, integrate over all space with the added condition :

Setting GenerateConditions to False may produce unexpected answers:

In this case, the condition that the integral is divergent was lost:

Interactive Examples  (1)

Consider Gabriel's horn, the interior of rotating around the axis for :

Compute the volume for arbitrary endpoint :

Compute the surface area for arbitrary endpoint :

The limit as of the volume is finite, but the surface area is infinite:

Visualize the horn along with its volume and surface area as functions of :

Neat Examples  (2)

The first six Borwein-type integrals are all exactly :

From the seventh onward, they differ from by small amounts, for example the eighth:

A logarithmic integral from Srinivasa Ramanujan's notebooks:

Wolfram Research (1988), Integrate, Wolfram Language function, https://reference.wolfram.com/language/ref/Integrate.html (updated 2019).

Text

Wolfram Research (1988), Integrate, Wolfram Language function, https://reference.wolfram.com/language/ref/Integrate.html (updated 2019).

CMS

Wolfram Language. 1988. "Integrate." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Integrate.html.

APA

Wolfram Language. (1988). Integrate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Integrate.html

BibTeX

@misc{reference.wolfram_2024_integrate, author="Wolfram Research", title="{Integrate}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Integrate.html}", note=[Accessed: 18-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_integrate, organization={Wolfram Research}, title={Integrate}, year={2019}, url={https://reference.wolfram.com/language/ref/Integrate.html}, note=[Accessed: 18-May-2024 ]}

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