Breaking down a numerical integration into two parts

Sorry for this stupic question.

Suppose we have this integral

Of couse this can be solved analytically. But I still want to solve it numerically as:

dx = 0.01;
x = 1:dx:4;
fx = x.^2 + 5*x;

I = trapz(x,fx)

I = 58.5001

Now, we break down this integral into two part. Analytically, we write it as:

I am confused about the lower limit of the second integral when solving it numerically. Should it be 2 or . If we use 2, will not it mean that we are counting one point two times, once in and once in ?

Answer 1

[Torsten]

I1 = dx/2 * f(x0) + dx * sum_{i=1}^{n-1} f(xi)     + dx/2 * f(xn)
I2 = dx/2 * f(xn) + dx * sum_{i=n+1}^{n+m-1} f(xi) + dx/2 * f(x_{n+m})
->
I1 + I2 = dx/2 * f(x0) + dx * sum_{i=1}^{n+m-1} f(xi) + dx/2 * f(x_{n+m})

Thus since the endpoints in the trapezoidal rule are only counted half, all is fine.

[Luqman Saleem]

Got it, thanks