Function Approximations Package 内置符号

# NIntegrateInterpolatingFunction

 NIntegrateInterpolatingFunction[f, {x, xmin, xmax}] gives a numerical approximation to an integral with InterpolatingFunction objects in the integrand. NIntegrateInterpolatingFunction[f, {x, xmin, xmax}, {y, ymin, ymax}, ...] gives a numerical approximation to a multidimensional integral.
• NIntegrateInterpolatingFunction uses the function NIntegrate, but it breaks up the domain of integration into sections where the objects are smooth.
• The arguments of the InterpolatingFunction objects may themselves be univariate functions of the integration variables.
• Numerically integrating a multidimensional integral using NIntegrateInterpolatingFunction with InterpolatingFunction objects containing a large number of nodes may take significantly longer than using NIntegrate.
• NIntegrateInterpolatingFunction has the same options as NIntegrate.
 例   (1)
Needs["FunctionApproximations`"]
A trapezoidal approximation to :
 Out[2]=
Since is not smooth, NIntegrate will generate a warning message:
 Out[3]=
Using NIntegrateInterpolatingFunction produces a slightly more accurate answer without any error messages:
 Out[4]=
In this case the integrand is simply an interpolating function, so you can use Integrate to check:
 Out[5]=
 Scope   (3)