IntegrationbyParts–TABULARMETHOD--PaulkNotesPage 1

SOURCE:http://pages.pacificcoast.net/~cazelais/187/tabular.pdf

IntegrationbyParts–TABULARMETHOD--PaulkNotesPage 2AdditionalcommentsbyKathyPaulk:

• Normally,wedecidewhichpartwillbeDIFFERENTIATEDbytheorder“LIATE”whichstandsfor:• Logs,InverseFunctions,Algebra,Trig,Exponents.Thishelpsustochoosethedifferentiatedpartthatismorelikelytoreducetozero.• IftheDsideisalogfunction ln 𝑥 ,thenwestopafteronestep !

!

• IftheDsidedoesnotreducetozero,thenwelookfortheproducttoaequalamultiple 𝑛 ∗ 𝐼 oftheoriginalintegral 𝐼 .• NOTE:HorizontalarrowèGiveup/STOPèTaketheintegraloftheproduct• NOTE:Giveup/STOPwhenDsidedoesnotreducetozeroororiginalintegralisamultipletheproductofD*I.(repeating)

LookingatExample#4inmoredetail:

• ThesourceoftheexampledidnotfollowtheLIATErecommendation.Instead,the“exponent”partwaschosentobedifferentiated.• Note,thatthetabularmethodworks,regardlessofwhichpartischosentobedifferentiated.

𝐼 = 𝑒!! cos 𝑥 𝑑𝑥

𝐼 = 𝑒!! (sin 𝑥 )− 2𝑒!! − cos 𝑥 + 4𝑒!! − cos 𝑥 𝑑𝑥

𝐼 = 𝑒!! (sin 𝑥 )+ 2𝑒!! cos 𝑥 − 4 4𝑒!! cos 𝑥 𝑑𝑥

𝐼 = 𝑒!! (sin 𝑥 )+ 2𝑒!! cos 𝑥 − 4 𝐼

5𝐼 = 𝑒!! (sin 𝑥 )+ 2𝑒!! cos 𝑥

𝐼 = !!!

!sin 𝑥 + 2 cos 𝑥

𝐼 = 𝑒!! cos 𝑥 𝑑𝑥

usingLIATErule… 𝐼 = cos 𝑥 !

!𝑒!! + sin 𝑥 !

!𝑒!! + (− cos 𝑥) !

! 𝑒!! 𝑑𝑥

𝐼 = cos 𝑥 !!𝑒!! + sin 𝑥 !

!𝑒!! − !

! cos 𝑥 𝑒!! 𝑑𝑥

𝐼 = cos 𝑥 !!𝑒!! + sin 𝑥 !

!𝑒!! − !

! 𝐼

!!

𝐼 = cos 𝑥 !!𝑒!! + sin 𝑥 !

!𝑒!!

𝐼 = !!

cos 𝑥 !!𝑒!! + !

! sin 𝑥 !

!𝑒!!

𝐼 = !!

cos 𝑥 𝑒!! + !!

sin 𝑥 𝑒!!

𝐼 = !!!

! 2 cos 𝑥 + sin 𝑥 Sameanswer!

D I

𝑒!! + cos𝑥

2𝑒!! − sin 𝑥

4𝑒!! + − cos 𝑥

D I

cos𝑥 + 𝑒!!

− sin 𝑥− !!𝑒!!

− cos𝑥 + !!𝑒!!