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 $\frac{dv}{dx}=1$
$$dv=1 dx$$
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