102week5 - Contents 1 Integration by Parts 1 1.1 The Rule...

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Contents1Integration by Parts11.1The Rule for Integration by Parts. . . . . . . . . . . . . . . . . . . . . . .11.2Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Further Integration82.1Table of Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.2Rational functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.2.1Long Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.2.2Completing the Square. . . . . . . . . . . . . . . . . . . . . . . . .112.2.3Partial fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .122.3Integrals of Functions Defined by Two-part Formulae. . . . . . . . . . . .152.4Odd and Even Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .163Curves Defined Parametrically183.1Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.2The Ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19i
iiCONTENTS3.3Arc Length in Parametric Form. . . . . . . . . . . . . . . . . . . . . . . .203.4Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203.5Area and Arc Length in Polar Coordinates. . . . . . . . . . . . . . . . . .223.5.1Arc length in Polar coordinates. . . . . . . . . . . . . . . . . . . .23
Chapter 1Integration by Parts1.1The Rule for Integration by PartsIfuandvare differentiable functions ofx, then the product rule yields:ddx(uv) =udvdx+vdudx.Henceuv+C=integraldisplayudvdxdx+integraldisplayvdudxdx,andintegraldisplayudvdxdx=uvintegraldisplayvdudxdxThis is the rule for integration by parts which applies to products. It does not solve theintegration problem but merely transforms it into a different integration problem.Thismethod is effective if one factor in the integrand can be easily integrated (without becomingtoo complicated) and the other factor can be easily differentiated. Candidates for factorsthat can be easily integrated areex, sinx, cosx, sinhx, coshx.Sometimes an artificialfactor 1 is introduced that integrates tox.Candidates for factors that simplify whendifferentiated are polynomials, but also lnx.In some cases the integration can be done by applying integration by parts severaltimes.1
2CHAPTER 1. INTEGRATION BY PARTSFor definite integrals integration by parts becomesintegraldisplaybaudvdxdx= [uv]baintegraldisplaybavdudxdx.1.2Examples(i) Findintegraldisplayxcosx dx.
(ii) Findintegraldisplayx2sinx dx.
1.2. EXAMPLES3

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