# What is the indefinite integral of zero? [duplicate]

From the definition of indefinite integral I might say: Since the derivative of a constant is zero, thus the indefinite integral of zero is a constant. Therefore: $$\frac{dc}{dx} = 0 \quad\iff\quad \int 0dx = c, \quad\forall c\in\mathbb{R}$$

However... we know that $0\in\mathbb{R}$, and since zero is a constant, I can pull it out the integral: $$\int 0dx = 0\cdot\int 1dx = 0\cdot(x+c) = 0$$

And then we end up that integral of zero is zero, not an arbitrary constant. Where is wrong here?