${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$udv$=uv-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vdu

We must first choose $u$ and $dv.$ We recall that we should do so in a way that ensures that du is simpler than $u,$ and $dv$ is easy to integrate, Keeping this in mind, for ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{4x}dx$ we can let $u=x$ and$=d{x}_1$