M408C - Notes - Integrals for Final - Contents 2 3 4 5 6 7 8 9 Exam Review Long Division and Complete-the-Square Example Finishing Off the

M408C - Notes - Integrals for Final - Contents 2 3 4 5 6 7...

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Contents2 Exam Review: Long Division and Complete-the-Square Example3 Finishing Off the Complete-the-Square Example4 The Integration by Parts example5 Which also involves trig sub6 Trig Integrals7 A Third Trig Integral8 Trig Sub9 The Last Trig Subbie
Exam Review: Long Division and Complete-the-Square ExampleHere it is, the exam review sheet. The topics to be covered are:a) Long division; completing the squareb) Integration by partsc) Trig Integralsd) Trig substitutione) Partial fractionsA)ExampleRx3+1x2+2x+3dx.When you have a quotient of polynomials, it’ll be either a completethe square problem or a partial fractions, but in either case you’llneed to do a long division first. ThenZx3+ 1x2+ 2x+ 3dx=Z(x-2) +x+ 7x2+ 2x+ 3dxWhat happens next depends whether the denominator factors ornot: wheretherb2-4ac0 orb2-4ac <0. Buta= 1, b= 2, c= 3,andb2-4ac= 22-4(1)(3)<0 so the denominator does not factor,and this is a complete-the-squares problem.Completing the square:x2+2x+3 = (x+e)2+f=x2+2ex+(e2+f),so ya got 2e= 2 ande2+f= 3. Thene= 1 and 1+f= 3 orf= 2.In all,x2+ 2x+ 3 = (x+ 1)2+ 2 andZx3+ 1x2+ 2x+ 3dx=Z(x-2) +x+ 7x2+ 2x+ 3dx=Z(x-2)dx+Zx+ 7(x+ 1)2+ 2dx=x22-2x+Zx+ 7(x+ 1)2+ 2dxNow do au-substitution in the integral:u=x+1;du=dx;x+7 =u+ 6. Then
Finishing Off the Complete-the-Square ExampleZx3+ 1x2+ 2

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