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 $0$.

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There are two types of integrals at play here.  Definite integrals
are the ones that describe the actual area under a curve.  Indefinite
integrals are the ones that describe the anti-derivative.
There's no paradox, really.  When speaking of indefinite integrals,
the integral of 0 is just 0 plus the usual arbitrary constant, i.e.,
$\int 0 \, dx = 0 + C = C $
There's no contradiction here.  When evaluating the area under a curve
f(x), we find the antiderivative F(x) and then evaluate from a to b:
$\int^{b}_{a} f(X) \, dx = F(b) - F(a)$
So, for f(x) = 0, we find F(x) = C, and so F(b) - F(a) = C - C = 0.
Thus, the total area is zero, as we expected.
