Study Guide on Integration

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Department
Mathematics
Course
MATA32H3
Professor
Szegadyand Grinell
Semester
Fall

Description
IndefiniteIntegrals SeptemberH27H10 1:23PM Integrationistheoppositeofdifferentiation. Whenfisgiven,thenfindF,thenF'=f Thisiscalledantidifferentiation Example *:6; L 6 á 6 Lë; I 6 L 6 - á=J(-2-O=J+)J)N=0 OK032-KJ K Note:Anygeneralsolutionsiscalledindefiniteintegral. IntegrationRules: 1. ±/(6 L /6 - ? I 2. ±6 (6 L 6Ô># - ?á= M .I = - I I 3. +*= L .Iá± (6 L ŽZ6 - ? 6 ë ë 4. ±) (6 L ) - ? 5. ±ŽZ6(6 L 6 Û ŽZ6 . 6 - ? 6. ±/*(6 L /± * (6 - ? 7. ± * / +(6 L ± *(6 / ±+(6 : ; % $ Eample:* 6 L M6 - 6 . I % $ M & I % ±M6 - 6 . I(6 L 6 - 6 . 6 - ? L K Eample:* 6 L I 6 ¾ # ±6 ?$(6 L J 6 - ? ¾ ë # % Eample: * 6 L ) - - 6 ë I : ; I I K ±) -ë - 6 - I ;% (6 L ) - ŽZ6 - 6 - 6 - 6 - 6 - ? 6 L J www.notesolution.com Example:* 6 L #>ë á6 M H ë I ± - I L ŽZ6 - 6 - ? 6 . Example:* 6 L ë >ë># ë># * 6 L 6 - I . I - I L 6 - I 6 - I 6 - I I I ±6 - (6L 6 - ŽZ 6 - I - ? 6 - I J www.notesolution.com IntegrationwithConditions NovemberH11H10 11:17AM Example:*ñ L 6 - Iá * H L J.FindF I * 6 L 6 - 6 - K J L I * 6 L 6 - 6 - J K Example:7 L * 6 ; ññ $ 7 L 6 7 H L M 7 I L I *6 Lë $ ñ I % ±6 (6 L 7 L 6 - K 7 L M L I % I & ± K - M(6 L 7 L IJ 6 - M6 - I 7 L I L - M - IJ L{ . IJ L # & &= 7 L #$ 6 - M6 . #$ ExampleofEconomicApplication: /" L JHHH . JHM . KM $ Lë $ $ % ±JHHH . JHM . KM (M L JHHHM . IHM . M - 7O3>O2-232-J+H L Má L H $ % Û M L JHHHM . IHM $ M L JHHH . IHM . M PhysicsApplication: f=Position,displacement f'=velocity f''=acceleration Givenf''=a=9.81 ñ ±=(2 L 4 L * L =2 - á L H-* *N)) *=00á>32 L 4 Ü www.notesolution.com I $ ±=2 - 4 Ü2 L ( L * L =2 J 4 2 - áÜ L H -J *N)) *=00á>32 L ( # I ( L 4 2Ü- =2 $ J www.notesolution.com MoreIntegrationTricks NovemberH11H10 11:39AM 1. Assumethatuisafunctionofx,thereforethepowerrule: Ô># Ô 3 ±3 (3 L = - I- Warning:Itmusthaveaduinthefunctionintegral ## #" :ŽZ6 ; ± :ŽZ6; M II - I (3 L (6á3 L ŽZ6 ;#" 6 #" ## :ŽZ6 ; :ŽZ6; ± 6 (6 L II - $" Example:ì - I ; (6 ? 3 L 6 - Iá(3 L I(6 $# $# $" 3 :6 - I; ±3 (3 L JI - L JI - Eample: ±6 6 - M(6 $ 3L 6 - Má(3 L 6(6 I # I # I J % I % ¥ :6 - M ;% ± 3 (3 L ±3 $(3 L Û 3 - L 3 - L - J J J K K K Eample: $ & $ ±L6 :6 - I ; (6 & % 3$L 6 - Iá(& L L6$(6 % ñ 3(3 L 6 - I ; Û L6 á-2(K)OJ 2 1=/) O)JO) 2K 3O) 2,-O 1)2,K(ä ±L6 $:6 - J6 - I (6; #" : & L ## z ; L ' ±L6 - z6 - L6 (6 L II6 - O - 6 -M è è 2. ±) (3 L ) - www.notesolution.com I 3. ± (3 L ŽZ3 - 3 Eample: ±) ë. Û 6(6 $ 3L 6 á(3 L J6(6 I I I . ± ) (3 L ) - L ) ë - J J J Eample: ±J #?ë(6 L ±) jl $:#?(6 3L ŽZJ I . 6 á(3 L .ŽZJ è I I è )jl $ #?ë ±) (3l FL ) - L - .ŽZJ .ŽZJ .ŽZJ Eample: 6 ± $ (6 6 - I $ 3 L6 - Iá(3 L J6 I I I I $ J ± 3(3 L J ŽZ3 - LJŽZ:6 -I; - Eample: I ± 6 . I (6 6 - I 6 . I ± . (6 :6 - I 6 . I ; :6 - I 6 . I ; I I I J ± :6 . I;. :6 - I; (6 I I ŽZ6 . I . 6 - I - J J ƒ’Ž‡ã I ± (6 6ŽZ6 I 3 L ŽZ6(3 L (6 6 I ± (3 L ŽZŽZ6 - 3 www.notesolution.com Example: I ± : ;%(6 ¾ ¾ 6 . J I 3 L ¾ . Já(3 L J¾6 J I ±3 ?%(3 L - L . - .J3 $ :¾6 . J;$ Example: ±IH 6 - K (6& 3 L 6 - Ká(3 L I(6 ±IH3 (3 L J3 - L J :6 - K;' - šƒ’Ž‡ã $ ' ±J6 6 - M (6; 3 L 6 - Má(3 L J6(6 ' I : I : $ ;: ±3 (3 L N 3 - L N 6 - M - Eample: & ' # ±L6 :JO - 6 ;%(6 ' & 3 L L - 6 á(3 L M6 (6 I ' # & L Û ± JO - 6 ; :M6 (6 ; M # & % & ± 3 (3 L L Û Û 3 - L 3 - L K :JO - 6 ';%- M M L M M Example: %ç>; ±M) (2 3 L K2 - O ( L K(2 M M M ± ) (3 L ) - L )%ç>; - K K K Example: www.notesolution.com ±6 ) &ë (6 & % 3L L6 á(3 L IN6 (6 I I ± ) (3 L ) &ë - IN IN Eample: IJ6 - L6 - J ± (6 6 - 6 - J6 % $ % 3 L 6 - 6 - J6 (3 L N6 - J6 - I(6 J(3 L IJ6 - L6 - J J $ % ± 3 (3 L JŽZ6 - 6 - J6 - Example: ññ I ñ 7 L 6 $á7 .J L Ká7 I L J I 7 L . - 6 I K L . .J - M L J ñ I M 7L . -6 J I M M ±. - (6 L .ŽZ 6 - 6 - ñ 6 J J M ñ J L .ŽZ I - IJ- I . L ?ñ J M I 7 L .ŽZ 6 - 6J. J Eample: ŽZ6 ± (6 6 I 3 L ŽZ6á(3 L (6 6 I I ±3(3 L 3 - L ŽZš ;$- J J ƒ’Ž‡ã I I ±J (6 L ±) jl $(6 L ) jl $- L J - ŽZJ ŽZJ www.notesolution.com Nte: I ±= ë(6 L = - ŽZ= Example: jl ë K ± 6 (6 I 3 L ŽZ6á(3 L (6 6 è I jl ë ±K (3 L K - ŽZK Example: ë. 6) ± ë . (6 ¥ ) - J 3 L ) ë. - J(3 L ) ë. Û J6 I ë. ± (3 L ¾ 3 - L ¥ ) - J - J ¾3 Eample: ±K ë jl(6 3 L 6ŽZ6 (3 L I - ŽZ6(6 ±K (3 I ë jl ë L K - ŽZK Integrationbyparts: :34 ;ñL 3 4 - 34 ñ ñ ñ ñ ± :34 (6 L ±3 4(6 - ±4 3(6 ±34 (6 L 34 . ±3 4(6ñ ±3(4 L 34 .±4(3 šƒ’Ž‡ã www.notesolution.com ±6) (6 3 L 6á(3 L (6 4 L ) - á(4 L ) (6 ±6) (6 L 6 ) - . ±) - (6 L 6 ) - .±) (6 .±(6 ë ë L 6 :ë - . :) ë;. :6 - ; L 6) - 6 . ) . . 6 . L 6) . ) - á5,)N) L . . L ) 6 . I - šƒ’Ž‡ã ±6ŽZ6(6
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