The formula of integration by parts is: $$\int u(x)v(x) dx = u(x)V(x) - \int u'(x)V(x) dx$$ Which can be re-written as: $$\int u(x)v(x) dx = u(x)[V(x)+C] - \int u'(x)[V(x)+C] dx$$ where C is a constant.
It makes some integration calculations simpler, such as:
$$\int x\tan^{-1}(x) dx$$ When we take $ u(x)=\tan^{-1}(x)$ and $v(x)=x .dx$, then $V(x)= \frac {x^2}2 + \frac 12$ instead of $V(x) = \frac {x^2}2$. It make steps calculations easier and simpler.
The question is: How to know and choose this constant? is there some guide or it just experience ?