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NIntegrate

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

NIntegrate[f, {x, xmin, xmax}]
    gives a numerical approximation to the integral of f with respect to x from xmin to xmax.

NIntegrate[f, {x, xmin, xmax}, {y, ymin, ymax}, ...] evaluates a multidimensional integral by adaptive cubature over a constant box, or iterated 1D quadrature when an inner bound depends on an outer variable. The variable is localised (HoldAll). xmin/xmax may be Infinity, -Infinity, or complex (a straight-line contour); extra nodes {x, x0, x1, ..., xk} give a piecewise-linear contour or mark interior singularities. Method -> Automatic chooses globally-adaptive Gauss-Kronrod for smooth finite integrands, double-exponential (tanh-sinh / sinh-sinh / exp-sinh) for endpoint singularities and infinite ranges and high precision, a Levin/zeros scheme for oscillatory integrands, an exponential endpoint map plus integration-between-the-zeros for an oscillatory endpoint singularity, and Monte-Carlo for high dimensions and region (Boole) integrands. Machine or arbitrary precision via WorkingPrecision.

Options: Method (Automatic | GlobalAdaptive | GaussKronrodRule | DoubleExponential | TrapezoidalRule | LevinRule | OscillatorySingularity | MonteCarlo | QuasiMonteCarlo | AdaptiveMonteCarlo | PrincipalValue), WorkingPrecision (default MachinePrecision), PrecisionGoal, AccuracyGoal, MaxRecursion, MinRecursion, MaxPoints, Exclusions.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= NIntegrate[Cos[x], {x, 0, Pi/2}]
Out[1]= 1.0

In[2]:= NIntegrate[Exp[-x^2], {x, 0, Infinity}]
Out[2]= 0.886227

In[3]:= NIntegrate[1/Sqrt[x], {x, 0, 1}]
Out[3]= 2.0

In[4]:= NIntegrate[Sin[x]/x, {x, 0, Infinity}]
Out[4]= 1.5708

In[5]:= NIntegrate[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
Out[5]= 3.14159

Implementation notes

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= NIntegrate[Sin[x], {x, 0, Pi}]
Out[1]= 2.0

In[1]:= NIntegrate[Exp[-x^2], {x, -Infinity, Infinity}]
Out[1]= 1.77245

In[1]:= NIntegrate[Exp[-x^2], {x, -Infinity, Infinity}, WorkingPrecision -> 30]
Out[1]= 1.772453850905516027298167483341

In[1]:= NIntegrate[Sin[x]/x, {x, 0, Infinity}]
Out[1]= 1.5708

In[1]:= NIntegrate[Log[x] Log[1 - x], {x, 0, 1}]
Out[1]= 0.355066

Notes

NIntegrate[f, {x, a, b}] approximates a definite integral. The Gaussian example reproduces Sqrt[Pi] = 1.77245..., computed to 30 digits with WorkingPrecision -> 30 via the double-exponential rule on the infinite range. The Dirichlet integral Sin[x]/x over [0, Infinity] is the oscillatory case, returning Pi/2. The final integral has the closed form 2 - Pi^2/6 = 0.355066.... Method -> Automatic selects globally-adaptive Gauss-Kronrod, double-exponential, Levin oscillatory, or Monte-Carlo schemes per region. Endpoints may be infinite or complex (a contour).