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Midterm

2006 Test 3 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137Y20062007WinterSessionSolutionstoTermTest31Evaluatethefollowingintegrals38isecxtanxdx1323secxtanxdxsecxsecxtanxdxsecxCbyasimplesubstitutionusecx31x8iix3dx0x3xIntegratingbypartsletuxdv3dxThendudxandvHenceln311xxxx11x3x333133xx3dxdx0222ln3ln3ln3ln3ln3ln3ln3000033ln32222ln3ln3ln3dx10iii42xx1LetxsecThendxsectandthuswehavesectantand3ddcosd3342secsectansecsec13Tointegratecoswetakeoutonepowerofcosandapplyasubstitution13223cosd1sincosdcossincosdsinsinC3x2thensinx1xHenceSubstitutingbackwecanusetrianglestoseethatifsec12322x1dxx11C342x3xxx1dx10iv32xx1Wedecomposetheintegrandusingpartialfractions1BEACD32232xx1xxxx1x1Obtainingacommondenominatorontherightsideandignoringthedenominatorgives2222331Axx1Bxx1Cx1Dxx1Ex432322433Ax2xxBx2xxCx2x1DxxEx432ADx2ABDExA2BCxB2CxCMatchingcoefcientsgivesusC1B2A3D3andE1Hence13213211dx3lnx3lnx1C2322xxxx1x1x2xx11
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