In this work sheet we'll study the technique of integration by parts. Recall that the basic formula looks like this:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$udv$=u*v-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vdu

$1$ First a warm$-$up problem. Consider the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xsin(3x)dx.$ Let $u=x$ and let $dv=sin(3x)dx.$ Compute $du$ by differentiating and $v$ by integrating, and use the basic formula to compute the original integral. Don't forget the arbitrary constant!

$2$ Compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}lnxdx.($The proper technique is$,$ indeed, integration by parts. What should you take to be $u$ and $dv?$ The choices are pretty limited$.$ Try one and see what happens.$)$