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Midterm

2001 Test 3 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137YTermTest3Solutions200120021Evaluatethefollowingintegralstt6ie1edtttLetu1eThenduedtthereforegivingus22tt3232te1eudu1eCudt335sinxdx6ii2cosxWhilethiscanbeintegratedbypartsitiseasiertorstrewritetheintegral5422sinxsinx1cosxdxsinxdxsinxdxI222cosxcosxcosxNowletucosxThendusinxdxgivingus22421u12uu12I2udududu222uuu331u1cosx2u2cosxCu3cosx32xx1dx7iii32x2xxWedecomposetherationalfunctionusingpartialfractions22xx1xx1ABC2232x2xxxx1xx1x1whereABCareconstantsSolvingfortheseconstants2222Ax1Bxx1Cxxx1ABx2ABCxAxx1MatchingcoefcientswehaveA1AB1and2ABC1SolvingwehaveB0andC1Hence2xx1111dxdxlnxC322x2xxxx1x131dx7iv22x2x21TheintegralcannotbedecomposedusingpartialfractionstheexpressionisalreadysimpliedinpartialfractionsCompletingthesquareofthequadraticinthenumeratorwehave3131dxdx2222x2x2x11112Letx1tanqThendxsecqdqsopp23133dxsecq2dqcosqdq422x11secq100pp331cos2q11dqqsin2q22400p12pp13p3sin643642681
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