Divide through by x and the left hand side is 'exact', i.e. the evident 'result' of a product rule differentiation.
Just in case a picture helps...
... is the product rule. Straight continuous lines differentiate downwards (anti-differentiate up) with respect to x.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!