RepeatedUse of Integrationby Parts
Oftentimeswe use integrationby partsmorethan onceto evaluatethe integral,as the examplebelowshows.
Example4:
Evaluate.
Solution:
With and our integralbecomes
As you can see, the integralhas becomeless complicatedthan the original,. This tells us
that we havemadethe right choice.However, to evaluate we still needto integrateby partswith
and Then and and
Actually, the methodthat we havejust usedworksfor any integralthat has the form , where
is a positiveinteger. The followingsectionillustratesa systematicway of solvingrepeatedintegrationsby
parts.
TabularIntegrationby Parts
Sometimes,we needto integrateby partsseveraltimes.This leadsto cumbersomecalculations.In situations
like theseit is best to organizeour calculationsto saveus a greatdeal of tediousworkand to avoidmaking
unpredictablemistakes.The examplebelowillustratesthe methodof tabularintegration.
Example5:
Evaluate.