How to find the arbitrary constant in an integral?

I guess you dont know what the word "arbitrary" means, then? Its precisely why we use a big C variable as opposed to some numeric value.

You need an initial condition. Let's say you have f(x) = x^2 and you want to know what the anti derivative of x^2 is for a certain initial condition. Let's say our initial condition is f'(0)=5. So when you evaluate the antiderivative at x=0 we know the AD is 5. Now... carry out the operation.
Integral: x^2 dx = (x^3)/3 + C. Now... f'(0)=5 so (0^3)/3 + C = 5 and that means C=5. So if you had a problem that said: f(x)=x^2 with f'(0)= 5... what is the arbitrary constant of integration... this is how you'd solved that problem. :) But... without an initial condition.. there's no way to determine it. :)

It depends of what is given in the question.
for example if you've found that the integral of a certain function is x² +3x³ +c
and in the question they give y(0)=1 , then you just put x=0 and equal it to 1:

0²+3*0³ +c =1

c=1

If you have just integrated and are trying to find the constant 'c'. You need to have a solution to the equation prior.

If the integral only includes x and y, you should be given a co-ordinate (x=X,y=Y). Now input X and Y into the equation and solve to find the constant 'c'.