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The integrator QAGS will handle a large class of definite
integrals.  For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
\int_0^1 x^{-1/2} log(x) dx = -4
The program below computes this integral to a relative accuracy bound of
1e-7.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>
double f (double x, void * params) {
  double alpha = *(double *) params;
  double f = log(alpha*x) / sqrt(x);
  return f;
}
int
main (void)
{
  gsl_integration_workspace * w 
    = gsl_integration_workspace_alloc (1000);
  
  double result, error;
  double expected = -4.0;
  double alpha = 1.0;
  gsl_function F;
  F.function = &f;
  F.params = α
  gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                        w, &result, &error); 
  printf ("result          = % .18f\n", result);
  printf ("exact result    = % .18f\n", expected);
  printf ("estimated error = % .18f\n", error);
  printf ("actual error    = % .18f\n", result - expected);
  printf ("intervals       = %zu\n", w->size);
  gsl_integration_workspace_free (w);
  return 0;
}
The results below show that the desired accuracy is achieved after 8 subdivisions.
$ ./a.out
result = -4.000000000000085265 exact result = -4.000000000000000000 estimated error = 0.000000000000135447 actual error = -0.000000000000085265 intervals = 8
In fact, the extrapolation procedure used by QAGS produces an
accuracy of almost twice as many digits.  The error estimate returned by
the extrapolation procedure is larger than the actual error, giving a
margin of safety of one order of magnitude.