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6.4 Completed Notes.pdf

# 6.4 Completed Notes.pdf - Sec 6.4 integration with TabIes I...

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Unformatted text preview: Sec 6.4 integration with TabIes I A. Tables TABLE OF INTEGRALS BASIC FORMS IW’ WW 0 nil v 2' n “+ 1 . n _ IE. I Ianudu = In Iscc "I + C u . II (In _ I I I3. I court!" = In |sinn| + C , ff ' 4 - I I4. I scour!" = In [sccu + 1mm} -I- C 1 = a . IS. lcscudu = In Icscu , cotuI + C , Ina . . rlu | 1—, 16. l—— = sin —-I- C l‘ _ . Ja’ H u’ (I I a2 _ I KW 'I‘ N I a I (It: I u . 13.:l—wu—H =Hscc"— + C 2 = 4. NW 0 (I - ' (In I u -I- a . art—m 19. | = —In + c 9 J 5c um 90 u _ a1 —- u’ 20 u -— a ' . ‘ In I u - (I I = - 20. | ‘ = —ln + c . , u2 - (1’ 2a r-I- u FORMS mvowmc J01 + "i. a > n ,.__.___ u 1 21. I ﬁll +u1riu=g~da1 + u? + Ez—lnﬁt + «W + u’) + C ' u ‘22. I “2 \lal -I- "2th: = Em? + 2:12) ‘10 1+ u’ — -—ln(u + \m! -|-u1)+ C O (:1 + u’ ‘ V‘a’ + u’ 23. Impr—du : 3/01 -|- u’ - a In ' «0’ -I- 1:3 Ja’ + u2 24. l -—2—du = ——-—~ -I- lnIu -|- ‘1‘“! + ”2) + C . u u ‘ (In 25. l —— = In(rr + \la’ + “1) -I- C I \faz -l- u’ ‘ u’du I_l 26. -—-— \ a’ --I H1 — —-Ill(u + Jul + u 2)"? . «1’ + “2 =2 2 the I a? -I- u" -I- (r 27. J——— = -—In I- c rula’ -I 1.:2 a H 28 l du __ \laz +111 + C I. u’v‘rr’ + "1 azu ' (In H 29. m — -I- C W: , .I (a + II‘I’” aka! +1.2 Desiré Taylor Math 1242 FORMS INVOWIRG \ (I’— 30. I \III- -— II- ’IIII m H :‘Iaz - II? + -— sin TABLE OF INTEGRALS II". «>0 ,._...__. I" — + C (l ‘ 4 . ,——— Ir a . __ II 3]. I N'Vrl': - “at!" = §'(2i'l'1 — (I'M/(I2 - II2 + ‘E—Slll ‘-~ + C (I | ' III2 — II’ (I + (I3 — N’- 32. I '\———‘—“(III' = \IIIJ — II1 - IIln + C . II II ‘ III’ - II-’ 1 II 33. I -\—-u:-—II'II = "—‘\I(I2 - II2 — sin' H+ C .. II' II (I ‘ ":6“! II (11 u 34. I I =-—\III3—II1-I—-sin'- +C . \I'II’ m II2 2 2 (I ‘ (III I II + III’ - II2 35.I ‘ 1=~h~In—~ﬁ+;—*— +6 . II \-'II- -- I'I' ﬂ ' IIII ] 1 . 36. I I, , =-z—'\III‘—II--I-(, . II \' II 7 II’ (I II ‘ : 1-5;: g H 2 2 2 2 3n 1 II 37. IIII “HI IIII—-—(2II —5(I) (I “H +——sin — -|-C ‘ 8 8 II (III II 38. I ﬂ = m + C [111)“ ﬂz\/ﬂ1—Ilz FORMS INVOLVING rI' (I > O . , I_I 39. I \ II3 — (I'I’III= 2 —'\II(2 - (I2 — ——lII III + III2 -" (I I -I- C . ' I I ‘ , II 40. I l'f'ffl" - II-IIII = EIQHI -— 02% III2 — (12'— ——lI1IIII \I! I3 —II1I+ C . ,’_‘—‘"1 J. v - - I - _ rI , 4?. I IIII = \IIIz - (I2 -- II cos '— + C . II |II| ' 5' 1h. 3 2 _ 1 I, II II \II'I' (I 4‘2. I , IIII = ——--—— + In III -I- \III2 — (I‘I + C . II' II " IIII 2 2 43. I 1%: E III III + \IH _. (l' I 'l' C . \-'II"— II- ' II’IIII II 44. I I‘_M‘=-— [Hz—(12+—'II‘IItI+\Il'I'3‘—‘(I‘ I+C . V‘H' - (I' (III \III7 — (I2 ‘ 45. I: -—— T = ‘—-—2— + C, . II \ "-'II w (r (I II IIII II 4" ”’ ‘ ﬁ—TMTW? 4“ C . (H * H I a 1/” - (I Desiré Taylor ____—-—-—I Math 1242 TABLE OF INTEGRALS FORMS INVOLVING u + bu ‘ u (lu I . : —~— + — + 47 'l u + bu b’ (u bu a In |u bu I) + C 48 l‘ H’dﬂ _[t +13): __ 4( +1] + 2 21+] ]+ C . . u + bu -2;’_ u u u u m) u n In M] ‘ du l u 49. m z u I + ‘ j u(u + bu) u n u + bu C ‘ du ! b u "I. bu . r,—— = ,2 + — 50 ‘ u'(u + bu) (m uz n u 5| ‘ uu'u _ u +[_l2| +1) I I-C ' _ (u + bu]2 b’(u i bu) n I“ H ‘ du I l u + bu 52' J u(u + bu)I _ u(u + bu) u F I“ u t + C 53 | "2!!” l a I bu “2 2 |1| +lr{ i C . '——"_'—,‘ ‘j‘ —‘ ' ' '_ "' 3 I' ‘ ' . {u -I~ bu)‘ b" u + bu a I u I 2 ‘ 54. l u\/u + bu (b — lib: + C I u du 5. —— I — 2 ‘l + b -i- 5 ‘ \ ,.__.aﬂ + bu 1’ b2 ()u u) u u C . 2 i 56. —/"_:_mﬁ = 1511‘ (Ba I- 3!) u — 4ubu)\/u + bu + C . u -' 57 l‘ du =~Ln Wail) "J5: +C ifu>0 - . u «a + bu J" la -1 bu + ' 2 u + bu ‘ = Ian"I + C, if < 0 «CH -u a ‘ J +1 ‘ I 58. ‘Mdu = 2 u + bu -I- (r l ML . H . u (a + bu 59 “—— -J__a" + bu Iu \lu + bu b"_ﬁ___| du . r = f u 2 um . 2 . 60. l u‘K/a 4- bu (be 3 m [u"(u + bu)”: - uu l u""' x a + bu du] 61 u "du _2u"\/u --I bu __ , 2m: ‘ u""‘du . . ‘11: + bu 119'"? I) M?“ + i). «u -|- bu “ (b: \m + bu b(2u — 3) ‘ (b: . u 62. W "Jr: 4- bu “(Pi — UM'H 2N0? # 1) . u""\/u + bu Desiré Taylor Math 1242 TABLE OF INTEGRALS TRIGONOMETHIC FORMS 63. I‘sinzu (I: = Eu — {-sin 2!! + C | 64. I 6052:: (hr = In -l~ isin 2H + C a 65. I Ian'utlu = [an N - u + C 66. I col’mhr = —co! u — u + C 67. I sin’rrdu = ~§(2 + sill’u) cos n + C 68. I cos’udrr = H2 + cos’n‘) sin n + C 69. I tan’udu = itnn’u -|- In Icos n| -f- C . 70. I col’udn = —§ collar ~- In Isin "I + C . ' I I II. I scc’ndu = j sec H mm: + 311: |scc r: -I‘ [an u| + C I I 7?. I csc’rrdn = "3 csc 1: cm H + ‘2!" Icsc u H cm “I + C ‘.,, l.,,__. u-l‘.,,_.. 73. I sm u m: = ——5m n cos u -I- sm 'mm . u n . . _ . d .n w. i n-'] ‘ H l n"! 7 . cos. (”In H cos :1 sum + cos “(In . n u . l 75. | WIN" # I Iml”'2udn u — . . I 1311"" (In = INVERSE TRIGONOMETRIC FORMS 87. I sin"urfrr = rrsin"u + JP '- "1 + C 1 33. I cos"'udu = ucos"'rr H \H - u’ 'l' C o v I mn"'u (In = ulan"u — ; |u(] + "1) + C M2 — l I In.” - u’ 4 sin" a -I- T- + C 89. 90 I u sin"u d" = ‘ __ 2M2 — 1 _ 91- I "cos 'rrdu =——cus ' ﬂ" Desiré Taylor 76. 77. 78. 79. 80. BI. 82. 83. 84. 85. 86. 92. 93. 94. 95. . *1 . I c0l”udu = cnt""u - I col""’udu _ n — I . ‘ l n — 2 ' F I scc"m!u = Ian rrscc""u + see" ’udu _ n — I u - l . ' —l n — 2 ‘ I csc"u(lu a col ucsc""’u + I csc""udu _ n - l u — l _ ' . . sin(n - b)u sink: + b)” . I smmr sm bu (In H m.— - — + C . 2(0 _ b) 20? + b) ‘ . . sm(n H 1))" sm((: + b)u I (:05 (m cos bud" = —~—-—-— —r- + C . 2(rr - b) 2({1 + b) ‘ . Cosh: - b)" cask! + b)" I sm (m cos bud" = —----—-— - —-—-— + C . 2(0 ~ b) 2(a + b] I "sin "(In = sin n - n cos n + C . . I ucosu (In = cos“ + H sill u + C I u"sin mm = -u"cos u + u I M" cos n du . . I n" cos n «In = u"sin l'l' — n I u""' sinudu . . ‘ sin“"'u cos'“'u n H l ‘ I sin"u cosmrr (m = u I sin""u cos'ﬁrdu . n + m n 1— m , sin"”u cos""u m - l ‘ . , = —— -I- I sm"u (205“ 2nd” n + m n + m . ‘ u2 + I 2 hurt" "1 ~52!- ‘I- C I u {allr'u (In = 1 [NH sin"'u - I u’”. d“ W] M \$5 —1 ' I I u" sin“! (In o n+l ‘ l ' u"” (In . It” 60541”!!! = — u”" c054“ + I — n ¢ —I ,I n -I- | . W ‘ ' _ l ' M” (In I u"lan 'mhr = -—-- ir"” lmf'n H I I . n 9E -I . u -f- l _ . I 'I' n Math 1242 TABLE OF INTEGRALS ___________________—..._._.___...__—.__——-_-—-———— EXPONENTIAI. AND EOGARITHMIC FORMS I l I 96. I Item'du = F01" -‘ l)e"" + C 100. I in min = r: In H ~ u + C Q ’ I H n “n+l 97. I “Regudu = _"ncau H .... ”n clued" 10]. I 1: mm!” m ——);[(n l- I) In H — 1] -l~c . a a , (31+ 98. I 0"" sin Land“ = 029+ ()2 (a sin bu ~ I) cos 1th) + C 101' H In H ‘1'”:l" II“ ”I + C V ell" I 99. I 9"" cos but!“ = 0’ + b2 ((1 cos bu + bsm bu) + C HYPERBGHC FORMS 103. inh n (In = cosh u -I- C IOB. I csch u (1:: = In |tanl1§u| + C 104. ech’n (in = tanh u + C 105. I tanh min = In cosh n -|- IIO. I csclg’udu = Heath u -l- C III. I scch H lanh u ch: = -scch u + C 106. I colhudrl = "I + C t a I csch l'l' cmh u the = —csch u + C 107 scch "dz: = tan" Isinh u] + C FORMS ENVOLVING film: h NE, a > 0 (l2 ”3. I «20:: — u (It! 3 u g a 2cm — “1 + "Ems '(a _ H) + C (I 6 2 a ‘ 2Hz — rm — 3a2 a3 (1 ~11 ”4. I raw/2m: — “3:!” = ———— anu — u2 + —cus" + C a — u «20:: - "2 + acos"'( ) + C __ 2 ”5' .’___.I ‘12:”: u du= a I" T 2' H. z .1 ”6. I‘___ 2m: n —-—du = _ 2‘1201: u _ cos"(ﬂ a u) + C l ‘ In a - u m. I—-‘—— = cOs"( )+ c . ‘12tm- "3 a I 18. I u du . J20" '- M2 ' 1 + 3 w n9. |ﬂ"= ”" ‘" 3“)\/27:;——ui+icos- I“ ") +c . ‘/2(m — u’ a “___r___I du J2rm - H2 u a — u = — 2(m - u2 + acos'( ) + C a =——H+c 120. ~./2(m — u? an Desiré Taylor Math 1242 Examples: 1.) [(25—4x2)%dx ,7: ‘4 ,4 a C \$1125 UL“: LIX! *7- ,‘2 (f% [Qu¢6w¢)m“ k 3%"Sw 71)"? c 5 “V'- 27‘ r’F—ﬁ—f—T‘T‘} a du." 20160 1T9; (’%(2[4X2)~5(25))\/25”4X +14%) 814,-. {g} ' +C Jiolu: 0W # _.___ :5: 241331115de "z 9‘ 3 9m; Sued/x ‘73 :3(”’._é-\$V;\“x cosx+ 5%.. SQVRBX ) W L 4; (p7 9(ﬁ’éL9“; qxcosx +751 (”*%(1 +3inzx)Cosx)) + 5;: t/f—b Desiré Taylor Math 1242 Desiré Taylor Math 1242 'f l ) We bwork I 2. (1 pt) t Book Problem 5 I dx Use the Table of hate ruls in the back of our textbook to evaluate f ——-—— g y x2 v 16X2 -i- 49 ,_ 11. %; AM '— LIL EJaej'L 10 .-;; F70 I "'—-—-_, Fm ”ﬁg; 4‘? (#7:) ‘ 7. (1 pt) Book Ploblem 15 ' .-— ,— H '1‘4be""+c Use the Table of Integrals in the back of yom textbook to evaluate / ﬁght 1'in X Desiré Taylor Math 1242 9. (1 pt) Book Problem 27 . . l7 ._ Evaluate the Integlal fx2+4x+29 (1A — ~_—; 11 gangs—l X2+fo+ 2 9 Desiré Taylor Math 1242 s. (1 pl) ) Book Problem 11 Use the Table of Integrals in the back of your textbook to evaluate fyM—l I + [By—y2 d)" W” (9 ,C-adsm Mw "‘ Cat-ﬂaw) (2) J7,— (€03- t-JE)“: (6&3? ~ (W) C33 m (kl—.3331 __ [Lg‘ywhzsj W .,-_ af‘v'uf" £152. 0M2” -2u.o(M -—-—~ ,Jliwr: ULOUL DesiréTaylor "Ll—S “W [f7- oh” 4, b ( 5:59 MW *E'SZ‘SVLHKﬂ'g’ 4‘: 77M .. Q. __________ . _, f ‘5 “if (25’£3“bl‘)m+ mean/26454.? +759m"'(.5_£5—9)K ...
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