11th - 8.5 TR“ }()N( “METRIC SU BSTITUTIUNS 11. y=7...

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Unformatted text preview: 8.5 TR“ }()N( “METRIC SU BSTITUTIUNS 11. y=7 secF),0 < 6‘ < %,dy =7secfitanfidf), Vyg — 49=7tan9; [WI—49 dy: = 7ftan39c16= 7f[sec36—1)c19 = mane — 91—C )1 Tsecé‘ =7[L”2““’—sec-1 111] —c 20. x = 2 51116, 0 4. F) 4. %, d1: = Enos 6 d6, (4 — x31“? 2 8 0033 6‘; 1-1 dx _ jinfi 2cos€d3 _ 'J_ mils d3 _ 1 [tan INTI _ fl _ 1 D (4_x2)i53 _ 0 Stem? _ 4 o cos-3 _ 4 1! _ l2. _ 4J3 32. y : emna, 0 < 6 < dy : em“? sec2 6 d6, (,1 l — (111 y 3 = 1,1 l — [£1112 : sec 6‘; -e d _ --.7_.-'4 “ml .2 a _ --.7_.-'4 _ “H4 _ 1 7—),‘1l‘t’lny13 — jg 2—1....1sszcca d9 — jfl sec 6 d9 — [1n |sec 6 — tan 6H”I — ln (1 — V6) 39. (113—41g23,c1y=3_$§;)1=3j;%=§1an—1 g—C;x=2anc1y=0 =1» ozgmn—H—C ::> C——%r 2:» Vzgtanq —% 8.0 INTEGRAL TABLES AND COMPUTER ALGEBRA SYSTEMS 3 dx 2 (12111—9 4] dx (1 I x'3\,1-'4x— 9 (—9k 13 xv'eix— 9 I (We used FORMULA 15 with a = 4, b = —9) _ 14—9 2 2 _ 14—9 —%—(Ej(fi)m]v‘e ‘C (We used FORMULA 13(11) with a = 4, b = 9) 1411—9 4 _ 14 —9 X — 31—9)‘ — 2—? tan V 9 — C ' 1131—4 _ ' d 29.j1t d1_2\,131—4—1—41]W3l+4 (We used FORMULA 12 witha: 3,1): —4) _ _ _ 1—__ 22(531—4—4(7231an 1V’3‘441—c24131—4—4m11 1 “I, 4—1:? (We used FORMULA 13(11) with a = 3, b = 4) 36. fsin sin % dt = 3 sin — sin — C (We used FORMULA 62(1)) with a = b = %) . _ “Zn/E . _ . 42.1%‘de; x2113 —. jmsulu-2ud11:21c05_]ud11:2(11c0s_]11—%\,11—113)—C \- dx=2udu (We used FORMULA 97 with a = l) = 2 («Rees—1 VE— V 1 — —C 71. fléx‘u‘flnfigdx:16["|fl:’d2—% '11311111c111] =16["‘“§’°2 —?3[#—13 fax“ (WeusedFORMULAllOwitha 1,11 3,111 Zanda l,n 3,111 1) t] 12 If] 1 -| 1 I 216(“M' —fl—%)—C=4X4(]11x)3—2x4lnx—%—C ' 2 -11 _x'=2-~_if sq _—x?2"_i _x2."‘_1_ .fl—x 7&sz dK_—ln2 1112 L dx— 1112 1112 1112 1112 - dx :_x?2—x_2[12—x 2—x]_c 1112 1n? —ln2 _ (1112]-2 (We used FORMULA 106 with a = — 1, b = 2, n = 2,11 :1) . _ t_ . _ . 81. je‘sec,“u‘[e‘—1]clt;[pi—e 1] —-jseeu‘xdx:%—%jsecxdx d11=et dt (We used FORMULA 92 with a = l, n = 3) 2 “Cam” — ln |sec x—tan x| —C = .13lsee[el — l] tan[el — 1] —ln |se{:[el — l] — tan[eI — 1]” — C 92. fcscw 2.x coth 2): dx = — “35;” — c = — my?" — C‘ (We used FORMULA 136 with a = 2,11 = 3) 8.7 NUIV'IICRICAI. INTEGRATION 5_ f2[t:+_t]dt l,- f(t,-) n1 1nf(t,-) D . b—a 2—0 2 1 t“ 0 0 l 0 I- (a) “"241sz n ZTZHZE u 112 513 2 514 =>%=%;me(tj)=25=>T=%(25)=%; 13 1 2 2 4 fit) 13—1 » 9(1) 313 _1 » f”(t) 61 1?, 312 39% 2 3914 0 fl 2:» M: 12: f”(2) => |ET| < % (.13]‘(12)— .13 ‘4 ‘ '0 l 10 (b) jo'tr" am: [if (i' if) 0:6 2 |ET|=jD'ui+—t]cu—T= —§_5=_% 3 |ET|=]E (c) ERIE >< 100: ><100:z4% 13. (a) M=2(see Exercise4): Thenng 2:» |ET| < %(§]3(2)= <10--1 2» n2 > @104] => 11 > Vguo‘l] ::> n > 115.4,soletn2 116 (b) M = 0(see Exercise 4): Thenn = 2 (n must be even) 2:» AK :1 2:» |Es| = % (1)4(0) = 0 < 10—4 8.8 IMPROPER INTEGRALS -b 4- In o [—mb— (—2151 12. lim [1n QR: lim (m b——1 —1n 24]j=1n(1)—ln(1§]=0—1n3=1n3 bdrm _ barn hr] 211 21. fieedfiz lim [Flee—e3]:=[0-e”—e”]— lim [beb—ebjz—l— 111111131] 13—1—09 b—1—oc- b—‘—'x_ : —] — b 21200 (_;_b] (l'H3pital‘s rule for form) = —] — 0: —l -:,-'2 I "J 36. j cot fidfi = lim [In |sin = ln 1 — lim [In |sin bl] = — lim [In |sin bl] = — so, 0 b —‘ 0* b —‘ D— b —t 0" the integral diverges .1 q .1 40. In dx; [5: = V/fl —. 2L 6—? cl}: = 2 — so the integral converges. 1 dx . - (L.-'x-1]_ - fl _ - 1 _ 1 _ dx _ - b_ 43- 4 WWI—'93” 1] —xl—'.“?x. W1 11mg. n+1 — 14—12161 4 w—bmc, [2w 4—90, \.-'x \- which diverges 2:» jd 7%1—1 diverges by the Limit Comparison Test. \- ...
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11th - 8.5 TR“ }()N( “METRIC SU BSTITUTIUNS 11. y=7...

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