# Numerical Integration

N[Integrate[expr,{x,x_{min},x_{max}}]] | try to perform an integral exactly, then find numerical approximations to the parts that remain |

NIntegrate[expr,{x,x_{min},x_{max}}] | find a numerical approximation to an integral |

NIntegrate[expr,{x,x_{min},x_{max}},{y,y_{min},y_{max}},...] |

| multidimensional numerical integral |

NIntegrate[expr,{x,x_{min},x_{1},x_{2},...,x_{max}}] |

| do a numerical integral along a line, starting at , going through the points , and ending at |

Numerical integration functions.

This finds a numerical approximation to the integral

.

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Here is the numerical value of the double integral

.

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An important feature of

NIntegrate is its ability to deal with functions that "blow up" at known points.

NIntegrate automatically checks for such problems at the endpoints of the integration region.

The function

blows up at

, but

NIntegrate still succeeds in getting the correct value for the integral.

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*Mathematica* can find the integral of

exactly.

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NIntegrate detects that the singularity in

at

is not integrable.

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NIntegrate automatically looks for singularities at the endpoints of the integration region and any subregions defined by piecewise functions (such as

Piecewise and

Abs) in the integrand. If additional singularities are present,

NIntegrate may not give you the right answer for the integral. Nevertheless, in following its adaptive procedure,

NIntegrate will often detect the presence of potentially singular behavior, and will warn you about it.

NIntegrate warns you of a possible problem due to the singularity in the middle of the integration region. The final result is numerically quite close to the correct answer.

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If you know that your integrand has singularities at particular points, you can explicitly tell

NIntegrate to deal with them.

NIntegrate integrates

expr from

to

, looking for possible singularities at each of the intermediate points

.

This gives the same integral, but now explicitly deals with the singularity at

.

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You can also use the list of intermediate points

in

NIntegrate to specify an integration contour to follow in the complex plane. The contour is taken to consist of a sequence of line segments, starting at

, going through each of the

, and ending at

.

This integrates

around a closed contour in the complex plane, going from

through the points

,

, and

, then back to

.

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The integral gives

, as expected from Cauchy's theorem.

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| | |

MinRecursion | 0 | minimum number of recursions for the integration method |

MaxRecursion | Automatic | maximum number of recursions for the integration method |

MaxPoints | Automatic | maximum total number of times to sample the integrand |

Special options for NIntegrate.

When

NIntegrate tries to evaluate a numerical integral, it samples the integrand at a sequence of points. If it finds that the integrand changes rapidly in a particular region, then it recursively takes more sample points in that region. The parameters

and

MaxRecursion specify the minimum and maximum number of recursions to use. Increasing the value of

guarantees that

NIntegrate will use a larger number of sample points.

and

MaxRecursion limit the number of sample points which

NIntegrate will ever try to use. Increasing

or

MaxRecursion will make

NIntegrate work more slowly.

With the default settings for all options,

NIntegrate misses the peak in

near

, and gives the wrong answer for the integral.

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With the option

,

NIntegrate samples enough points that it notices the peak around

. With the default setting of

MaxRecursion, however,

NIntegrate cannot use enough sample points to be able to expect an accurate answer.

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Another way to solve the problem is to make

NIntegrate break the integration region into several pieces, with a small piece that explicitly covers the neighborhood of the peak.

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For integrals in many dimensions, it can take a long time for

NIntegrate to get a precise answer. However, by setting the option

, you can tell

NIntegrate to give you just a rough estimate, sampling the integrand only a limited number of times.

Here is a way to get a rough estimate for an integral that takes a long time to compute.

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