Integration by Parts \u4e2d\u65b9.pptx - TECHNIQUES OF INTEGRATION In this chapter we develop techniques for using these basic integration formulas to obtain

Integration by Parts u4e2du65b9.pptx - TECHNIQUES OF...

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TECHNIQUES OFINTEGRATIONIn this chapter we develop techniques for using these basic integration formulas to obtain indefinite integrals of more complicated functions.
Integration By PartsStart with the product rule:This is the Integration by Parts formula.ddvduuvuvdxdxdxd uvu dvv dud uvv duu dvu dvd uvv duu dvd uvv duu dvd uvv duu dvuvv du
The Integration by Parts formula is a “product rule” for integration.udifferentiates to zero (usually(.dvis easy to integrate.Choose uin this order: LIPETLogs, Inverse trig, Polynomial, Exponential, Trigu dvuvv du
INTEGRATION BY PARTSformula for integration by parts.ExampleFindExampleFindduvvudvudxexxdxxxsinxudxedvxdxduxevxuxdxdvsindxduxvcosdxexedxexxxx
Example:logarithmicfactorLIPETlnx dxlnux1dudxxdvdxvxu dvuvv dulnxxxC1lnx xxdxx u vv du
Example 6:LIPETcosxex dxu vv ducosxex dx2cossincosxxxex dxexexsincoscos2xxxexexex dxCsinsinxxexx e dxxuesindvx dxxduedxcosvxxuecosdvx dxxduedxsinvxsincoscosxxxexexex dxsincoscosxxxexexx edx
A Shortcut: Tabular IntegrationTabular integration works for integrals of the form:where:Differentiates to zero in several steps.Integrates repeatedly. fx g x dx
INTEGRATION BY PARTSExampleFindREMARK2: in some integral, we may need to apply integration by parts many times.dtett22tte2t20tetetediff2tudtedvttdtdu2tevduvvudvudtteetdtetttt222tu2dtedvtdtdu2tevdtetedttettt222Ceteetdtettttt)22(22
INTEGRATION BY PARTSExampleFindREMARK2: in some integral, we may need to apply integration by parts many times.ExampleFindExampleFinddtett22tte2t20tetetediffdxexx23dxxxsin2
INTEGRATION BY PARTSformula for integration by parts.ExampleFindREMARK3: sometimes a repeated application of integration by parts leads back to an integral similar to our original one. If so, this expression can be combined with original integral.

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