Then
Substitutingbackinto the formulato integrate,we get
As you can see, this integralis worsethan whatwe startedwith! This tells us that we havemadethe wrong
choiceand we mustchange(in this caseswitch)our choicesof and
Remember, the goal of the integrationby partsis to start with an integralin the form that is hard to
integratedirectlyand changeit to an integral that lookseasierto evaluate.However, hereis a
generalguidethat you may find helpful:
- Choose to be the morecomplicatedportionof the integrandthat fits a basicintegrationformula.
Choose to be the remainingterm in the integrand. - Choose to be the portionof the integrandwhosederivativeis simplerthan Choose to be the
remainingterm.
Example2:
Evaluate
Solution:
Again,we use the formula.
Let us choose
and