EXAMPLE 3 Find r2e3rd.SOLUTHON Notice that t2 becomes simpler when differentiated (whereas e3t is mostly unchanged mhen sifferentiated or istegrated), so me chosse u=t2,dv=e3tdt Then of=0ofr=Infegration by parts givesr2e3rdr=-23e2rdrThe integral that we oteained, fe3H et, is simpler than the original integral but is sull mot obvious. Therefore, we une integration by parts a second time, this Elme with u=t and dv=e3t et. Then dv=dt,v=, andPutting this in Equation (3), we getr2e3rdr=13r3e3-=13r2e3r-(13re3r-19e3rc) C1where c1=-23c.