Numerical Integration using Simpson's 3/8 Rule

Simpson's 3/8 rule is a popular method to numerically evaluate the definite integral of a mathematical function f(x). It is a special case of the Newton-Cotes curve fitting formula where any given function is broken down into pieces of equal finite width(let's call this width 'n'). The function is broken in such a way that the number of pieces is a multiple of 3. In other words, n is a multiple of 3. So, if we take the simplest case where n = 3, there will be 4 points where the function's value is known. A cubic polynomial is to be fitted between those 4 points. The area under this curve is an approximation to the area under the curve f(x) in that piece. As the value of n increases, the approximation gets better and better and eventually, it will tend to the actual value of the integral.

The function simpson38 takes up 3 input arguments(a function handle, lower limit, and upper limit) and the 4th one(number of pieces) is optional. By default, it will take n as 60 pieces.

It returns the value of the definite integral as the output.

Example:

Inputs:
>>f = @(x) 1 / (1 + x);
>>I = simpson38(f , 0 , 1 , 18)

Output:

>>I = 0.6931