What is the integral of 0? I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.

Indefinite integrals (anti-derivatives) are known modulo a constant function. With definite integrals, the case is different:
$$
\int_a^b0\,\mathrm{d}t=0
$$
One way to verify that $C$ is the anti-derivative of $0$ is simply
$$
\frac{\mathrm{d}}{\mathrm{d}t}C=0
$$
