The rule holdsfor Whathappensin the casewherewe havea powerfunctionto
integratewith say
. We can see that the rule doesnot worksinceit wouldresultin divisionby.
However, if we posethe problemas finding suchthat , we recallthat the derivativeof
logarithmfunctionshad this form.In particular,. Hence
In additionto logarithmfunctions,we recallthat the basicexponentionalfunction, was special
in that its derivativewas equalto itself.Hencewe have
Againwe couldeasilyprovethis resultby differentiatingthe right side of the equationabove.The actual
proofis left as an exerciseto the student.
As with differentiation,we can developseveralrulesfor dealingwith a finitenumberof integrablefunctions.
Theyare statedas follows:
If and are integrablefunctions,and is a constant,then