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] can be entered as f x.
can be entered as int or \[Integral].
is not an ordinary d; it is entered as dd or \[DifferentialD].
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 any expression. However, Integrate uses only its literal form. The object , for example, is not converted to .
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:
Integrate can evaluate essentially all indefinite integrals and most definite integrals listed in standard books of tables.
In StandardForm, Integrate[f, x] is output as f x.
See The Mathematica Book: Section 1.4.4, Section 1.5.3 and Section 3.5.6.
Implementation Notes: see section A.9.5.
See also: NIntegrate, DSolve, Sum, LaplaceTransform, FourierTransform.