//自動選取步長梯形法算法
	
#include<iostream.h>
#include<math.h>

double fun(double);
void main()
{	
	double a,b,h,e,T0,T1,S;
	int n;

	cout<<"**************自動選取步長梯形法算法**************"<<endl;
	cout<<"請輸入積分區間的[a,b]:"<<endl;
	cout<<"a= ";		cin>>a;
	cout<<"b= ";		cin>>b;

	cout<<"請輸入容許誤差e:"<<endl;
	cout<<"e= ";		cin>>e;

	h=(b-a)/2;
	T1=(fun(a)+fun(b))*h;
	n=1;

	int flag=1;
	while(flag==1)
	{	T0=T1;
		S=0;
		for (int k=1;k<=n;k++)
		{
			S+=fun(a+(2*k-1)*(h/n));
		}
		T1=T0/2+S*(h/n);
		if (fabs(T1-T0)<3*e) 
		{	flag=0;
			cout<<"所求函數f(t)=2/(1+t*t)在["<<a<<","<<b<<"]的積分近似值為:"<<T1<<endl;
		}
		else {	n=2*n;	flag=1;	}	
	}
}
double fun(double x)
{
	return 2/(1+x*x);
}