%% Approximates the Integral of (5x^2-24x-12)/(x^3-4x)  
%% on the interval [3,5]. 
clc
format compact 
format long g
% Defining polynomials for the numerator and denominator
p= [5 -24 -12]
q = [1 0 -4 0]
rp = roots(p) % Not really important
rq = roots(q)
[R,P,K] = residue(p,q) % Calculates the partial fraction decomposition
% Begins integrals
a=3, b=5
n1=10, n2=1000
dx1=(b-a)/n1 
dx2=(b-a)/n2
% Constructs a list of midpoints
x1=[a+dx1/2:dx1:b];
x2=[a+dx2/2:dx2:b];
% Plugs the values into the rational function
y1=polyval(p,x1)./polyval(q,x1);
y2=polyval(p,x2)./polyval(q,x2);
sum1 = sum(y1)*dx1
sum2 = sum(y2)*dx2
exact = log(7^7/(3^8*5^4))