Find the derivativeof
Solution:
We use the QuotientRule and the naturallogarithmrule:
IntegralsInvolvingNaturalLogarithmicFunction
In the last section,we have learnedthat the derivativeof is. The antiderivative
is
If the argumentof the naturallogarithmis then thus
Example6:
Evaluate
Solution:
In general,wheneveryou encounteran integralwith an integrandas a rationalfunction,it mightbe possible
that it can be integratedwith the rule of naturallogarithm.To do so, determinethe derivativeof the denom-
inator. If it is the numeratoritself,then the integrationis simplythe \ln of the absolutevalueof the denomi-
nator. Let’s test this technique.
Noticethat the derivativeof the denominatoris 1, whichis equalto the numerator. Thusthe solutionis
simplythe naturallogarithmof the absolutevalueof the denominator: