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Код

#include <stdio.h>
#include <math.h>

int maxloop = 1000;

double int_func(double x)
{
    return 5*x*x+7*x+12;
}

double calc(double (*func)(double), double a, double b, double step)
{
    double _a = a;
    double _b = a + step;
    double res = 0;
    while(_b <= b)
    {
        double x = (_a + _b)/2.0;
        res += func(x)*(_b-_a);
        
        _a = _b;
        _b += step;
    }
    return res;
}

double solve(double (*func)(double), double a, double b, double eps)
{    
    double r1 = 0,
        r2 = 0;
    
    int loop = 0;
    
    double step1 = (b - a),
        step2 = (b - a)/2.0;
    
    do
    {
        loop++;
            
        r1 = calc(func,a,b,step1);
        r2 = calc(func,a,b,step2);
        
        if (loop > maxloop)
        {
            printf("Слишком много проходов\n");
        }
        
        step1 = step2;
        step2 = step2/2.0;
    }
    while(fabs(r1 - r2) >= eps);
        
    return r2;
}

int main(int argc, char* argv[])
{
    double eps = 0.1;
    double a = -1;
    double b = 5;
    
    double res = solve(int_func,a,b,eps);
    
    printf("Ответ: %f\n",res);
    
    return 0;
}