I want to write the code for solving these indefinite integrals, and then approximate it with boundaries by Montecarlo method. Thanks for help.
MATLAB: Code for the integrals in Matlab!
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Do you have the symbolic toolbox?
If so, start with the int command. You can do both indefinite and definite integration.
syms x realf = 5 + 1.0./sqrt(1-x^2) % dot-divide not strictly needed here
Q = int(f,x) % indefinite
Q = int(f,x,0,1/2)Now that we have an anlytical answer, 
, we can validate numerical schemes.
, we can validate numerical schemes. First we should make a vectorised anonymous function of the integrand,
f = @(x) 5 + 1.0./sqrt(1-x.^2) Matlab has pre-canned routines for the final two integration schemes:
A3 =quadl(f,0,0.5)A4=integral(f,0,0.5) % Global Adaptive Quadrature
For the (composite) trapz and Simpson's you need to decide on a step size (as you did above).
h = 0.1; % step size (guess)
x=0:h:1/2; A1= trapz(x, f(x))A quick hack of the Simpson's rule is
n = 10; % must be even
a = 0; b = 0.5; % integration limits
x = linspace(a,b,n+1)';c = [1,repmat([4 2],1,n/2-1),4,1];A2 = (b-a)*c*f(x)/n/3; %
It might be prudent to check all numerical values with the analytical one, especially the Simpson's implementation above.
No. You cannot use a numerical integration tool (integral) to solve indefinite integrals, and certainly not symbolic ones.
Not all problems have solutions. The art of mathematics is to find a way to solve intractable problems, instead of throwing them at a computer and hoping the computer can solve it for you. Computers are great at solving messy problems that are tractable, if only you had a vast amount of time and patience.
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