W kontekście rozmaitych forumowych "specjalistów" od metod numerycznych -
zabawne, jak tym razem będą wyjaśniać trywialny w istocie rezultat...

exact 0.5022749400837604
tapez 0.5022749194207212
simpson 0.5022738634614975

/*
* tq.c
* 2012 (C) slawek
*/

#include <stdio.h>
#include <math.h>

#define NPTS 10000 /* NPTS must be even! */

double x[NPTS+1];
double y[NPTS+1];

double f(double x)
{
return sin(x)*exp(-x);
}

double integrate_exact(double a, double b)
{
return ((cos(a) + sin(a))*exp(-a) - (cos(b) + sin(b))*exp(-b))/2.0;
}

double generate_points(double x[], double y[], double a, double b, int n)
{
int i;

for(i = 1; i <= n; i++)
{
x[i] = a + (i-1) * (b - a)/(n - 1);
y[i] = f(x[i]);
}
}

double integrate_trapez(double x[], double y[], int n)
{
int i;
double s = 0.;

for(i = 2; i <= n; i++)
s += (y[i] + y[i-1]) * (x[i] - x[i-1]) * 0.5; // an naive approach

return s;
}

double integrate_simpson(double x[], double y[], int n)
{
int i;
double odd = 0., even = 0.;

for(i = 1 ; i < n ; i+=2) odd += y[i];
for(i = 2 ; i <= n ; i+=2) even += y[i];

return (2.0*odd-y[1]-y[n] + 4.0*even)*(x[3]-x[1])/6.0;
}

int main(int argc, char* argv[])
{
const double a = 0.;
const double b = 5.;

generate_points(x,y,a,b,NPTS);
printf("exact %.16lg\n", integrate_exact(a,b));
printf("tapez %.16lg\n", integrate_trapez(x,y,NPTS));
printf("simpson %.16lg\n", integrate_simpson(x,y,NPTS));

puts("press enter...");
getchar();
return 0;
}