CK12 Calculus - Single Variable

Solution:
Whilewe couldeasilyintegratethe right side and then differentiate,the FundamentalTheoremenablesus
to find the answervery routinely.

This applicationof the FundamentalTheorembecomesmoreimportantas we encounterfunctionsthat may
be moredifficult to integratesuchas the followingexample.
Example5:
Use the FundamentalTheoremto find the derivativeof the followingfunction:

Solution:
In this example,the integralis moredifficult to evaluate.The FundamentalTheoremenablesus to find the
answerroutinely.

LessonSummary

1. We usedthe FundamentalTheoremof Calculusto evaluatedefiniteintegrals.

FundamentalTheoremof Calculus

Let be continuouson the closedinterval

1. If function is definedby , on then on


2. If is any antiderivativeof on then

We first note that we havealreadyprovenpart 2 as Theorem4.1.

Proofof Part 1.

1. Consider on