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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, xmin, xmax}] can be entered with xmin as a subscript and xmax as a superscript to .
• Multiple integrals use a variant of the standard iterator notation. The first variable given corresponds to the outermost integral, and is done last.  »
• Integrate can evaluate integrals of rational functions. It can also evaluate integrals that involve exponential, logarithmic, trigonometric and inverse trigonometric functions, so long as the result comes out in terms of the same set of functions.
• Integrate can give results in terms of many special functions.
• Integrate carries out some simplifications on integrals it cannot explicitly do.
• You can get a numerical result by applying N to a definite integral.  »
• You can assign values to patterns involving Integrate to give results for new classes of integrals.
• The integration variable can be a construct such as x[i], or any expression whose head is not a mathematical function.
• For indefinite integrals, Integrate tries to find results that are correct for almost all values of parameters.
• For definite integrals, the following options can be given:
 Assumptions \$Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate answers that involve conditions on parameters PrincipalValue False whether to find Cauchy principal values
• Integrate can evaluate essentially all indefinite integrals and most definite integrals listed in standard books of tables.
Indefinite integral:
Definite integral:
Use Esc int Esc to enter and Esc dd Esc to enter :
Use Ctrl+_ to enter the lower limit, then Ctrl+% for the upper limit:
Multiple integral with x integration outermost:
Integrals that may not converge are by default returned as conditionals:
Indefinite integral:
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Definite integral:
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Use Esc int Esc to enter and Esc dd Esc to enter :
 Out[1]=

Use Ctrl+_ to enter the lower limit, then Ctrl+% for the upper limit:
 Out[1]=

Multiple integral with x integration outermost:
 Out[1]=
 Out[2]=

Integrals that may not converge are by default returned as conditionals:
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 Scope   (22)
Rational functions can always be integrated in closed form:
Sometimes they involve sums of Root objects:
Similar integrals can lead to functions of different kinds:
Many integrals can be done only in terms of special functions:
Integrate special functions:
Integrate piecewise functions:
Multiple indefinite integrals:
Integrate an interpolating function:
Rational functions:
Algebraic functions:
Elementary functions:
Special functions:
Piecewise functions:
When there are parameters, conditions are often needed:
Integrate generalized functions:
Integrate a constant over a unit disk:
More general integral over a unit disk:
Regions can be given as logical combinations of inequalities:
Visualize 2D regions using RegionPlot:
Regions can be in any dimension; in this case integrate over a cone:
Visualize 3D regions using RegionPlot3D:
Integrate a function with parameters, getting a piecewise result:
Infinite number of regions:
Integrals involving unknown functions are done when possible:
Symbolic integrals can be differentiated with respect to parameters:
The variable of integration need not be a single symbol:
Combination of indefinite and definite integration:
Integrand with a removable singularity:
 Options   (5)
With no Assumptions, conditions are generated:
With Assumptions a result valid under the given assumptions is given:
Specify assumptions to evaluate a piecewise indefinite integral:
Generate a result without conditions that is valid only for some values of parameters:
The ordinary Riemann definite integral is divergent:
The Cauchy principal value integral is finite:
 Applications   (6)
Area of a disk with radius r:
Intersection of a disk intersected with a square region:
Find the volume of a 4-dimensional unit sphere:
Mean and variance of the normal distribution:
Average distance from the origin to a random point in the unit square:
Compare to the asymptotic result:
Construct Enneper's minimal surface using Weierstrassian integrals:
Evaluate integrals numerically using N:
Assumptions about variables can yield different forms:
Solve a simple differential equation:
Derivative with a negative integer order does integrals:
Laplace transform of an integral:
 Possible Issues   (13)
Many simple integrals cannot be evaluated in terms of standard mathematical functions:
Simple-looking integrals can give complicated results:
The derivative of an integral may not come out in the same form as the original function:
Different forms of the same integrand can give integrals that differ by constants of integration:
Results for integrals can depend on the way parameters appear in an integrand:
The indefinite integral of a continuous function can be discontinuous:
Parameters like are assumed to be generic inside indefinite integrals:
In definite integration conditions are generated:
The integration variable cannot itself be a mathematical function:
When part of a sum cannot be integrated explicitly, the whole sum will stay unintegrated:
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:
Expanding RootSum objects from integrals may give large results:
A definite integral may have a closed form only over an infinite interval:
With too many components in a piecewise integral, \$MaxPiecewiseCases may have to be increased:
Borwein-type integral:
A logarithmic integral from Srinivasa Ramanujan's notebooks:
Nested piecewise integrals: