NIntegrate

• Multiple integrals use a variant of the standard iterator notation. The first variable given corresponds to the outermost integral, and is done last.

• NIntegrate by default tests for singularities at the boundaries of the integration region, and at the boundaries of regions specified by settings for the Exclusions option.

• NIntegrate[f, {x, x₀, x₁, ..., x_k}] tests for singularities in a one-dimensional integral at each of the intermediate points x_i. If there are no singularities, the result is equivalent to an integral from x₀ to x_k. You can use complex numbers x_i to specify an integration contour in the complex plane.

• The following options can be given:

• NIntegrate usually uses adaptive algorithms, which recursively subdivide the integration region as needed. MinRecursion specifies the minimum number of recursive subdivisions to try. MaxRecursion gives the maximum number.

• NIntegrate usually continues doing subdivisions until the error estimate it gets implies that the final result achieves either the AccuracyGoal or the PrecisionGoal specified.

• For low-dimensional integrals, the default setting for PrecisionGoal is related to WorkingPrecision. For high-dimensional integrals, it is typically taken to be a fixed value, usually 2.

• You should realize that with sufficiently pathological functions, the algorithms used by NIntegrate can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NIntegrate.

• Possible explicit settings for the Method option include:

• With Method->{"strategy", Method->"rule"} or Method->{"strategy", Method->{rule₁, rule₂, ...}} possible strategy methods include:

• Methods used as rules include:

• Additional method suboptions can be given in the form Method->{..., opts}.

• The method suboption "SymbolicProcessing" specifies the maximum number of seconds for which to attempt performing symbolic analysis of the integrand.

• N[Integrate[...]] calls NIntegrate for integrals that cannot be done symbolically.

• NIntegrate first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.

• NIntegrate has attribute HoldAll, and effectively uses Block to localize variables.