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slides19 - MAT1300 Lecture 18 Improper Integrals Pieter...

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Unformatted text preview: MAT1300 Lecture 18 Improper Integrals Pieter Hofstra November 24, 2009 Overview Integration by Parts Improper Integrals 1 Integration by Parts 2 Improper Integrals Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Integration by parts Recall from last time the formula Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Integration by parts Recall from last time the formula Z uv d x = uv- Z u v d x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 xe- 5 x d x . Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 xe- 5 x d x . Solution. We let u = 3 x , v = e- 5 x . Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 xe- 5 x d x . Solution. We let u = 3 x , v = e- 5 x .Then u = 3 v =- 1 5 e 5 x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 xe- 5 x d x . Solution. We let u = 3 x , v = e- 5 x .Then u = 3 v =- 1 5 e 5 x Now we can use the integration by parts formula: Z uv d x = uv- Z u v d x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 xe- 5 x d x . Solution. We let u = 3 x , v = e- 5 x .Then u = 3 v =- 1 5 e 5 x Now we can use the integration by parts formula: Z uv d x = uv- Z u v d x Z 3 xe- 5 x d x = 3 x- 1 5 e- 5 x- Z (3) · - 1 5 e- 5 x d x =- 3 5 xe- 5 x + 3 25 e- 5 x + C Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Solution. Set u = ln( x 2 ) , v = 3 Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Solution. Set u = ln( x 2 ) , v = 3 which gives us u = 2 x x 2 = 2 x v = 3 x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Solution. Set u = ln( x 2 ) , v = 3 which gives us u = 2 x x 2 = 2 x v = 3 x So integration by parts gives us Z uv d x = uv- Z u v d x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Solution. Set u = ln( x 2 ) , v = 3 which gives us u = 2 x x 2 = 2 x v = 3 x So integration by parts gives us Z uv d x = uv- Z u v d x Z 3 ln( x 2 ) d x = 3 x ln( x 2 )- Z 2 x · 3 x d x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R 3 ln( x 2 ) d x . Solution. Set u = ln( x 2 ) , v = 3 which gives us u = 2 x x 2 = 2 x v = 3 x So integration by parts gives us Z uv d x = uv- Z u v d x Z 3 ln( x 2 ) d x = 3 x ln( x 2 )- Z 2 x · 3 x d x = 3 x ln( x 2 )- Z 6 d x Pieter Hofstra MAT1300 Lecture 18 Overview Integration by Parts Improper Integrals Example Evaluate R...
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slides19 - MAT1300 Lecture 18 Improper Integrals Pieter...

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