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...

• 58

This preview shows page 1 - 11 out of 58 pages.

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.