The integration of dx/dt respect to t is equal to x + c. What is the variable c?

Question

why $\int \frac{dx}{dt} dt = x + c$?

what is the variable $c$?

I think the constant $c$ is ommited, as like $\frac{d(x + c)}{dt} = \frac{dx}{dt}$.
Is my assumption correct?

Answer 1

When you are solving an indefinite integral $\int fdx$ for any real function $f:\mathbb R\rightarrow \mathbb R$, you are essentially solving the differential equation $g'(x)=f$ for $g$. Say you have found a solution $g$, then $g+c$ for any real number $c$ is also solves the differential equation as $(g+c)'=g'=f$. It follows that $c$ can be any real number.