Nowmakea changeof variable Choose Then or
Substituting,
This integralshouldlook obviousto you. The integrandis the derivativeof the naturallogarithm
Anotherway of writingit, since , is
UsingSubstitutionon DefiniteIntegrals
Example6:
Evaluate
Solution:
Let Then or Beforewe substitute,we needto determinethe new
limitsof integrationin termsof the variable.To do so, we simplysubstitutethe limitsof integrationinto
:
Lowerlimit: For
Upperlimit: For
We now substitute and the associatedlimitsinto the integral: