A trapezoidal approximation to :

Since is not smooth, NIntegrate will generate a warning message:

Using NIntegrateInterpolatingFunction produces a slightly more accurate answer without any error messages:

In this case the integrand is simply an interpolating function, so you can use Integrate to check:

NIntegrateInterpolatingFunction threads element-wise over the first argument:

Complex-valued interpolation:

Multidimensional integrals:

The arguments of the interpolating function may themselves be univariate functions of the integration variables:

A trapezoidal approximation to Sin[π x]:

Accumulate the sampling points used by NIntegrateInterpolatingFunction:

Plot the sampling points. The function is sampled at the x coordinates in the order of the y coordinates:

Accumulate the sampling points used by NIntegrate:

With NIntegrate, the nonsmooth behavior of f[x] near the points x= produces an error message and requires many recursive steps to evaluate accurately:

Increasing the order of the interpolation will produce a smoother function:

With a smoother function, fewer function evaluations are needed by NIntegrate:

If the interpolation is smooth enough, NIntegrate will require fewer function evaluations than NIntegrateInterpolatingFunction:

Multidimensional interpolating functions with a large number of nodes may take much longer to integrate using NIntegrateInterpolatingFunction instead of NIntegrate:

With NIntegrate, only one integral is evaluated, but the nonsmooth behavior generates many recursive steps:

Using NIntegrateInterpolatingFunction, the integral is broken up into integrals over a smaller domain, where the integrand is smooth: