The Natural Log Function: IntegrationLog Rule for IntegrationTry It OutFinding AreaUsing Long Division Before IntegratingSlide 6Change of VariablesIntegrals of Trig FunctionsAssignmentThe Natural Log Function: IntegrationLesson 5.7Log Rule for Integration•BecauseThen we know that•And in general, when u is a differentiable function in x: 1ln( )dxdx x=1lndx x Cx= +�1lndu u Cu= +�Try It Out•Consider these . . .233xdxx-�2sectanxdxx�Finding Area•Given•Determine the area under the curve on the interval [2, 4]2lnyx x=�Using Long Division Before Integrating•Use of the log rule is often in disguised form•Do the division on this integrand and alter it's appearance22 7 32x xdxx+ --�22 11 Remainder 192 2 7 3xx x x+- + -Using Long Division Before Integrating•Calculator also can be used•Now take the integral192 112x dxx� �+ +� �-� ��Change of Variables•ConsiderThen u = x – 1 and du = dxBut x = u + 1 and x – 2 = u – 1 •So we haveFinish the integration( )( )321x xdxx� --�( ) ( )31 1u uduu+ -�Integrals of Trig Functions•Note the table of integrals, pg 357•Use these to do integrals involving trig functionstan 5 dq q�( )10sin cost t dt+�Assignment•Assignment 5.7•Page 358•Exercises 1 – 37 odd 69, 71,