integration by parts

start with $\int (f\cdot g{)}^{\prime }dx$
using product rule for derivatives, $=\int (f^{\prime }\cdot g)dx+\int (f\cdot g^{\prime })dx$

left side simplifies to $f\cdot g$

isolate one derivative with $\int f\cdot g^{\prime }dx=f\cdot g-\int f^{\prime }\cdot gdx$

which gives $\int u\cdot v^{\prime }dx=uv-\int u^{\prime }\cdot vdx$
where $u=f$, $v=g$