NIntegrate
 Usage
Notes
 Further Examples
One Dimension NIntegrate will often give the numerical value of a definite integral without parameters faster than finding the definite integral and then using N.
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Functions that cannot be integrated using algebraic routines can nonetheless be integrated numerically.
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NIntegrate's default option values sometimes lead to a wrong answer.
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Here are the options of NIntegrate. The default value of Automatic corresponds to GaussKronrod in one dimension and to MultiDimensional in several dimensions.
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Increasing the number of recursive bisections by setting the option MaxRecursion gives a more reliable result.
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The previous result agrees with that obtained by Integrate.
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Here is a function that NIntegrate suspects has a singularity.
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Here is the plot of ff.
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NIntegrate does the integration correctly.
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This calculates the integral numerically to different precision goals, and accumulates the sampling points used for the integral approximations with the option EvaluationMonitor.
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This plots the points that were accumulated in samplePoints. The function is sampled at the x coordinates in the order of the y coordinates.
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Here is a function that oscillates rapidly.
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Here is its plot.
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Using NIntegrate with its default options settings gives an incorrect result.
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NIntegrate gives the correct result using  .
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It happens that an exact result is possible and taking N verifies the previous result.
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Multidimensional Low-dimensional integration can be done either with genuine space integration rules or with a Cartesian product of one dimensional rules. We compare the two using the very simple two-dimensional function  . Here are the sample points that come from using MultiDimensional, the default for NIntegrate.
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Here are the sample points from using the Cartesian product setting GaussKronrod.
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Here we will use NIntegrate on a more complicated function. Here is its plot.
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Here is a plot of the sample points for the numerical integration using  .
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For comparison, here is a plot of the sample points for the numerical integration using  .
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MonteCarlo  is occasionally advantageous in dimensions greater than  .
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Setting the option Method to MonteCarlo[24] is equivalent to using SeedRandom[24] to reset the random number generator.
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You can specify how many function evaluations are used in the methods MonteCarlo and QuasiMonteCarlo by giving a value to MaxPoints (the default being  ).
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If the AccuracyGoal and PrecisionGoal are set to Automatic, they will be set to  in the methods MonteCarlo and QuasiMonteCarlo, and to  in all other methods.
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