Math 1007, Fall 2006(These slides replace neither the text book nor the lectures.)Part V: Integration & Applications37. Antiderivatives (p. 110-117)38. Sigma Notation (118-119)39. Area Under a Curve and Riemann Sums (120-127)40. Properties of the Definite Integral (128-129)41. The Fundamental Theorem of Calculus (130-135)42. Indefinite Integrals (136-137)43. Integration by Substitution (138-147)44. Integration by Substitution for Definite Integrals(148-159)45. Using symmetry to simplify integrals (160-162)46. Areas between curves (163-182)47. Integration by Parts (183-194)48. Integration by Parts for definite integrals (195-196)49a. Integration by Partial Fractions: General Set Up(197-199)49b. Integration by Partial Fractions: Case 1 (200-203)50. Trigonometric Integrals (204-210)51. Trigonometric Substitution (211-216)52. Volume (217-220)53. Overall Strategy for Integration (221-222)Not covered in Fall 2006:Integration by Partial Fractions (Case 2,3,4)Volume by cylindrical shells109
37. AntiderivativesDefinition:A functionFis called anantiderivativeoffon an intervalIifF(x) =f(x) for allxinI.Examples:F(x) =x44is an antiderivative off(x) =x3sinceF(x) =4x34=x3=f(x)F(x) = lnxis an antiderivative off(x) =1xsinceF(x) =1x=f(x)F(x) = tanxis an antiderivative off(x) = sec2xsinceF(x) = sec2x=f(x)F(x) = sinxis an antiderivative off(x) = cosxsinceF(x) = cosx=f(x)Note thatddx(sinx+ 5) = cosx. HenceG(x) = sinx+ 5is also an antiderivative off(x) = cosx.In fact, for any constantc,H(x) = sinx+cis anantiderivative off(x) = cosx.110
Theorem:IfFis an antiderivative offon an intervalI, then the most general antiderivative offonIisF(x) +cwherecis an arbitrary constant.Theorem:Suppose thatFandGare both antideriva-tives offon an intervaleI. ThenG(x) =F(x) +cfor some constantc.Definition:f(x)dxis called an indefinite integral off.f(x)dx=F(x)meansF(x) =f(x)Findingf(x)dx=F(x) means finding themost generalantiderivative off.111
Examples (over 2 pages)1·dx=x+ck dx=k x+c2·dx=x2+cx4dx=x55+cxndx=xn+1n+ 1+c,n=-11xdx= ln|x|+cexdx=ex+caxdx=axlna+c112
cosx dx= sinx+csinx dx=-cosx+csec2x dx= tanx+csecxtanx dx= secx+ccscxcotx dx=-cscx+ccsc2x dx=-cotx+c113
We can find aparticularantiderivative by adding oneor more condition that will fix the constant.Example 1:Iff(x) = 12x2-24x+ 1 andf(1) =-2,findf(x).
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