calculus - Integration by parts: integrating both sides

Integration by parts is defined by the formula $\int udv=uv-\int vdu$. Let's says $\int udv=\int lnxdx$. When determining what $v$ equals, I learned that this requires integrating both sides of $dv=1dx$.
$$\int dv=\int 1dx$$
$$v+C=x+K$$
Are the constants $C$ and $K$ going to be equal to each other or different? $v+C=x+K$ doesn't even seem to carry any meaning because one antiderivative is in terms of $v$ while the other is in terms of $x$. My biggest confusion is integrating both sides of an equation with respect to different variables. It doesn't make sense to me.

Answer

$$v=x+(K-C)$$

Both K and C are real arbitarry constants so $K-C$ also must be a real constant.

Secondly, The equation you got just a relationship between $x$ and $v$

For example, $v=x+10$

differentiate both sides you get $\frac{dv}{dx}=1$

$$dv=1 dx$$