$\int\!{f \left( x \right) \frac{ dg \left( x \right)}{dx}} \ \mathrm{d} x = \begin{vmatrix} u = \frac{f\left( x \right)}{dx}, dv=dg\left( x \right) dx \\ du=\frac{df\left( x \right)}{dx} dx, v=g\left( x \right) \end{vmatrix} = \frac {f \left( x \right) g \left( x \right)}{ dx } - \int\!{g\left( x \right) \frac {df\left( x \right)}{dx}} \ \mathrm{d} x$
I'm a little bit confused by this $dx$ denominator.