'Half Bounded' Integrals?

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I'm just starting to learn integrals, and I know that indefinite integrals usually have some constant C at the end (something like f(x) + C). This makes sense to me, but I was wondering if you had a 'half bounded' integral so to say, for example the integral of x^2 from 0 to x - how would you solve for that. I know you must end up with some function without C (or it is possible to solve for C), but how do you do it? Thanks!