7.1 Notes.pdf - W “W W M I ““le ‘fl'xw It(a.1 m\I 1...

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Unformatted text preview: W “W W M I ““le ‘fl'xw It (a. . .1 m \I 1 Cl \NQ aim \r\‘\'lgro\tQ.o\ XUALJVIOM 4‘ WQ wafi *' r. “we llxoflv I3 0m affirm‘k cull 4*“ O‘RA‘M‘Q‘ N 7.1—lntegration’by Parts “+9 We oflm ortantlnte rols Pd‘bQ (4:4 4 @4qu chIXIdx= cjflxldx 44[f(x)+g(x)]dx = 4f(x)dx+4g(x)dx 7 Xn+1 :‘l ] _ _, andx= +c Inst-” 4 j—dx—Inlxl+C/,I n+1 x . _, .1 A Jede= e +C Ibde= b +C Inb Isinxdx=- -cosx+C Icosxdx=sinx+C J‘sec2 xdx= tanx+C Jcsczxdx=-cotx+C secxtanxdx = secx + C Jcscxcotxdx = ~cscx + C 43inhxdx= coshx+C 4coshxdx=sinhx+C 4tanxdx= ln|secx4+C 4cotxdx=ln4sinx4+C ——dx=—tan +C =sin +C,a>0 0 l2) Ifi—M will , . . ‘ . T. 2’: Review from MTH 141. The Substitution RuleKRQ/ MUM RH“ 3414 hw\ J . )) 0 Many integrals need to be adjusted so you can antidifferentiate using a rule. With substitution, othe idea IS to temporarily write the integral in terms of another variable, u, so that an antiderivative rules can be applied. . If U: g(x) IS a differentiable function whose range is an interval l and f IS continuous on I, then: 4f(g(x))g'(x)dx =_4f(u)du Exam les ot the Substitution Rule I. jx+5l7dx : WA 5 Ital. ; (M MQJMQ, SKX‘l‘BP AX “ T : .Efi/ +Q v :4 1: Marina W. x :3 4 C] or \ X 2.JX2X32dx:S~/\ Xv WwngéLA m: >< ... l : U\ z. x911“; 1.13 , '7”: ,L WW2 3 X 3cos(x + 2L+ Xw—wmm -.....1 M Wm, ;:'W' W“ Lot III: 56* +1 CM?— 4‘)? <tI>< J PIMUI. :Ce X 0\\< “i m ”t 0% : 2i I“ I :3...“ :{IS COS‘JI )wl : qwflx +1) I“ C \ The Substitution Rule for Definite Int WWWWWMJ Should a definite integral require substitution, you have two options: 1. Return to x: Complete the substitution process, integrate, return to x (or the original variable), use FTC Part 2. 2. Stay' m u: Complete the substitution, change the limits of integration to u— values, integrate, use the FTC Part2. >( \ LII“: {AI t\ (3in 2y 0\ \\—\9M\<£°\“’;‘ Q54 E gle M . V‘MKWU‘ ““w’ I I» (<36 lxam kefi hr ”‘1an \ k 1}: l3?” K19 egg?" F i W‘ ‘\ om?! WW 3*“de 7«: I «:9 fl: I 7K: xxx: fibl Return to x\ Z Stay in u 2 VIIUEXMT‘: 1» M M SW3“? : Abbe“ luv :1 III“) 2— 1/4 ‘1 ’2, Ft I l *QQWOIK ._ 21W ‘ a " « ,. ng \ AZEXAB Ii) :JZKLLVZI}: EZK Qgtg’“ :47qu xi‘t S X ' 0 ML LU’W o . ~ 'U'V‘Voi, M. Tmfis 01$ \r\3I LSFG\‘\'\W-\ :0 KW WM; \ Marino { W‘Nflffi W d\t’“\\w \ékl r WNMVK KO“ pk fiygem 19 \ «(lemme C”\ A Q WWW 0&5 AN: 28:..be WIDIIr‘flV 1R+Q%FII‘V\ TQLIVIIIIQLIQS \O‘J‘IIIS {In \erqngIIm IN §W\35‘(I\u‘\{§y\ I SIIIIVQ\Q§IWIQIIII \Mm Ifl WQ hII IIu 7 lnte rationb Pa heorem :Of \IIKL", 3gb) IRWIN!» W05“ IMO/IX . Integration usin substitution IS based on reversing the chain rulefo differentiation. %/0\’\ (,(kn, LRQGK “MAI Wflfk bd\ (QQ-(ILII‘ICIVIXV‘I 'K5 0 Integrat by s s b sed on re§b¥ving the product rule for differentiation: pr «9:: \+§:(UV) =t9<rjlv + $33 Kl“ iA‘I {Of-I’ve I0“ MN I.“ Hi? 0Q Lur r IQ WAIV: 601}; W!“ ImI W‘ILL (\rIIQILI d d d \m QCEA‘IXO FRI: LII IMO: LZIIRIIII’ (Ixuw‘) ~\JI\I E’s/\IngI-IIQ/ (IX-\WN hFIQCQQN w,Id(uv)= uv+C IIIIWIIL I) mum) +" \) W7 “ d d 2%» II \‘ I" L I \IIM III, 5V: UV JV U @515: “Dim (\ INIQA‘IMAQ “ H EIAQCJ 0be \Is \er tr :30 Item (Jefj‘ 0\ IL QIQIIM MI: \R‘VLSIIS / ({Ifixfif Useful Guidelines' In Choosin u a d \dv ‘ I AggQQXIg. Let u and v be differentiable functions of x. The Iu dv = uv — v du (XML U)‘ (“A cw) , M mQ/‘WJP \u \ SQ" 0 Let dv be the most complicated factor of the int grand that can be III IQ I0\ II B integrated easily. "vative IS a simpler function ”Wm! MWKNWI 0 Let u be a factor of the integrand whose WWWWMW m “M... 0 Use the LIATE method for choosing u. (Q m MI E _ | Inverse Trigonometric Function Ck V“ a R V . \\ A Algebraic Function E Exponential Function (N QIUWVIIMQ“ 0“? VII uyfi-IL Wwfi If there IS a product of two of these types of functions In an integral, choose u to \Q 3 1w L Logarithmic Function I I :3 y: EM KI T Trigonometric Function m; .I M " be the type that appears first In LIATE. Then dv IS whatever IS left. _ . x 0' IN} WRQIWQ Imfiééflw W (\Qw ‘9er I Imam? \I bI’I 30% SWEEVNNExamM les oflnte rationb Parts RAW Q_:€EIQKI\‘ Q. “2”“ ”Wk 95-39”- W W3 Ixexdx "' _._ “WW W” \\7\ \3 ¢\IQI‘IM\IIJ ~ \ I“ I I ;:\Lb-\\bu 9%} “fig §\I\ qux UL)?“ X doI§¥gé €ng VS III Iofilk I L Mtg} {KIM l “W ‘N 30?; 3°13???“ WI 35:13? 1:333 III) m \\ K Ic (IVS? LI (“\KJ cgmfimfl “VIIIIWQJVQINNIVQ NI ; (MI ‘E‘V 3‘ I \IV \\\ \_«;Q \r\ ~ Sham II (\3 \IIIW" “SQ/\QQH \3 I P \30‘N‘W‘7Q \& “mg" “m” "QT-\w X‘m-w w “W. “WW “memmuwwm I . \m m I , . Mg: 34 CM: SIAIIV 30953“ "‘ V \ “W \JV‘vaIVI If It QING— (II \I: gsxmxI‘I; : 4,63% LXMQI_I3IIXIM ._, “MW““MMM‘M‘” I Marino w, M’ X Lwa 4‘ 3 \Page 20 AFC/(I RI MI IIEIIQ UNI/LI: \I amyw INJWLQ X1 x dud/(Mas M351 “gm X ‘1 ‘ LL“ ~\ \ : ”:NX IV” ~Z+\ K Integration by Paris — More Than Once I. Ixzesxdx \‘XZ’Rxf, GAQSLBMMK \\ By” [«\<\"‘\ \3 (3%me M & LUV \F 342” \I: LBxKM \ K (M: MM \ll‘T-Szf’dx 15 ’ HIV \P 3% $9: E.) "‘ M : L . St «‘5 59,. H 7;“ \3 a 31mm \‘ 5w \5 (DLQOAQIW‘M //ME\ excosxdx ... S /3\ wt! 3\\ 3333\3 “”vi V33 3 \ Lipommw q )4 , X: (33 if \Kguskufl 533‘W330W03W‘3f‘ 3g _ Q, U353- \- 53% 334A“! 33 54 . 3W3 3:“:ng (M 9’ M“ s3 ‘ Qxflusy + 33:433me 5‘ 33—— 33 $333 3 M W, ,,,,,,W,_,____A_,_ & 3WQXQQAQ 3373333 \f\«3agr~o\3\olx 1x3 33’ “:3th ‘3 3rq§in3mt~3flb 3904" ES CXSGAVA 3LL3 \3 "SW CM“: LX354: 3, 33311— msfibk \3 “ITEM 3, L. ., 3333353982336 \r’\\3g 3 \X 3\\3 ¢ UM! m. 3 \3 030‘ E3 \ Q, c0584 +EJSVW< “”835 (“333% 3 ()(‘uij 33 33 £053,333: QUSX ‘33 SW34 ~« 3 <1, 3335x333 Lau‘X‘ova S 33 £333.33 mmmmwngwm”NW-...,,W...M,,,.Wm.m...m-mmmi-mw mmhmm M mm mm“ iii Q LL‘ " "'““ _, 3 (M "‘5‘ 6/ “’5 5L 3 3 “"54 ‘3' L WM 3 ,, “KEEN“..MMU N 39 0“ 333333333 7:, ~My.“m..mm,,_,.._.,W...u.,”WWWW,WWII;hm...“....m...,,,,,,,_,fi_mW“WWW W. m 3339:9333 30 _,,., \ ‘i\ X (W33 ~_. ’\ , F“ x Q/ Log X. “T Q SV‘QL 3 3L \ mfg)” orlno '; Definite Integrals by Paris b b Judv = [UV]: —I vdu 1/17,}?HV’ 1 Jxe*dx O “K” \5 Q\§5&.E)ra\g x m -x b" ‘XQ/ “ Q4 U m A A ’ V \ w w “[9 »~\ Exam le of Definite Infe ralsb Parts ...
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