Integrating with respect to time: $dx/dt=a$

Question

If $a$ is a constant and $dx/dt=a$, then integrating with respect to time gives $x = x_0 + at$. How do we show that?

Is my reasoning correct?

$dx/dt=a \implies \int_{0}^{t} dx(t) = a\int_{0}^{t} dt \implies x(t) - x(0) = at \implies x = x_0 + at$