Evaluatingthe secondintegralon the right,
Addingthe two results,
Remark:In the previousexample,we split the integralat However, we couldhavesplit the integral
at any valueof withoutaffectingthe convergenceor divergenceof the integral.The choiceis com-
pletelyarbitrary. This is a famousthoeremthat we will not provehere.That is,
Integrandswith InfiniteDiscontinuities
This is anothertype of integralthat ariseswhenthe integrandhas a verticalasymptote(an infinitedisconti-
nuity)at the limit of integrationor at somepointin the intervalof integration.Recallfrom Chapter5 in the
Lessonon DefiniteIntegralsthat in orderfor the function to be integrable,it mustbe boundedon the in-
terval Otherwise,the functionis not integrableand thus doesnot exist.For example,the integral
developsan infinitediscontinuityat becausethe integrandapproachesinfinityat this point.However,
it is continuouson the two intervals and Lookingat the integralmorecarefully, we may split
the interval and integratebetweenthosetwo intervalsto see if the integralconverges.