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Integration Techniques Worksheet. Prove properties of logarithms and exponential functions using integrals. h�\Rak�0�+��]��� ���6�~�2c2j@[i����r�:;�5w���K�)���T���d���ݛ�~��~鍷�1 �b��� ��"��6��0���j�>��D�����������)g;Su�$Κ�O&���&���ID��=6fo�l]8��uO��7;�)�jg��ɚOWW��ͱ! %���� ReportLab Generated PDF document http://www.reportlab.com 24 0 R ] 25 0 obj For example, f(x)=3x is an exponential function, and g(x)=(4 17) x is an exponential function. /Width 357 >> 25 0 obj endobj /FormXob.d7477e114638f2c471ec4a55453def9f 12 0 R Exponential functions occur frequently in physical sciences, so it can be very helpful to be able to integrate them. /ColorSpace /DeviceRGB /ColorSpace /DeviceRGB << /BitsPerComponent 8 % 'FormXob.9beb123a9d65cff433a6c59c3520a4fe': class PDFImageXObject % 'FormXob.714905f6842e063ca4ffcc2eec80bee2': class PDFImageXObject 0 View Math147 _lesson 5 Integration of Exponential Function.pdf from MATH 147 at Mapúa Institute of Technology. endobj /Subtype /Link /Subtype /Image /Subtype /Image /Width 233 >> /Height 93 % 'Annot.NUMBER6': class PDFDictionary /ColorSpace /DeviceRGB << /BitsPerComponent 8 25 0 obj /FormXob.d0ad280a0e6d51989bdc78d7f1afc991 17 0 R /FormXob.dce2fe057584215fb743584ad2229fb1 22 0 R % 'FormXob.51afb6fcee6da4fb19fc583658f41bb1': class PDFImageXObject /Width 1020 >> 10 0 obj << /BitsPerComponent 8 /Height 93 4 0 obj 21 0 obj /URI (http://en.wikipedia.org/w/index.php?title=Error_function) >> /ColorSpace /DeviceRGB *Qendstream 3 0 R << /BitsPerComponent 8 << /BitsPerComponent 8 /Type /Annot >> The function f(x) = 1x is just the constant function f(x) = 1. 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730.9469 % 'FormXob.aee9e3f2812d3133c4af185558af2733': class PDFImageXObject 17 0 obj d eae dx = Integrating the exponential function, of course, has the opposite effect: it divides by the constant in the exponent: 1 edx e ax ax , a /FlateDecode ] Exponential functions can be integrated, and you can test your ability to do so with this quiz and worksheet combo. /Width 1020 >> stream /Type /XObject /ColorSpace /DeviceRGB endobj Calc. /Type /XObject Anal. stream endobj /FlateDecode ] << /BitsPerComponent 8 /FlateDecode ] /Width 1283 >> % 'FormXob.39fed13b54d6df9e6ff552768a41b9e5': class PDFImageXObject /Subtype /Link endobj << /BitsPerComponent 8 /Height 93 /Subtype /Image The exponential integrals , , , , , , and are defined for all complex values of the parameter and the variable .The function is an analytical functions of and over the whole complex ‐ and ‐planes excluding the branch cut on the ‐plane. �}^�p��w�{��y� stream << /F1+0 70 0 R /FormXob.4caa680fc779b56803f6127fe3f16f9f 21 0 R /Width 630 >> /Filter [ /ASCII85Decode /FormXob.dce2fe057584215fb743584ad2229fb1 22 0 R /Width 1293 >> endobj /Length 7806 )Sdtq"I!7(#Uo/!8u0Wn.2endstream /FormXob.714905f6842e063ca4ffcc2eec80bee2 16 0 R /FlateDecode ] /Type /XObject /Type /XObject /FlateDecode ] 19 0 obj % 'FormXob.706a4adf6192e00b64f6556e0a16ed22': class PDFImageXObject /Subtype /Image Such a function is called an exponential function. /FormXob.6b55703c672f52a33c5d64d8ba7d1bb0 8 0 R "Lj89#.k?6^~>endstream % 'FormXob.f41b62a04397ad4481fe1f06a74ba49e': class PDFImageXObject /FormXob.b1efc86d94e5bcf517ee6576c537d13b 7 0 R /ColorSpace /DeviceRGB << /BitsPerComponent 8 endobj We’ve shown that differentiating the exponential function just multiplies it by the constant in the exponent, that is to say, d d x e a x = a e a x. /FormXob.39fed13b54d6df9e6ff552768a41b9e5 10 0 R We will assume knowledge of the following well-known differentiation formulas : , where , and , where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a. /Length 4026 /Height 100 /Width 423 >> 20 0 obj Applications of the Complex Exponential Integral By Murían S. Corrington 1. /Contents 88 0 R /Filter [ /ASCII85Decode Gb"0TgMS8e$p`5:s,V&W1aPR^"3/uJ11-IplC"4E!65h.#7j,&\rl17"?cfi;2d(E.DS?"TV@.JI_nr!!Nu0O';U7Jd?`PLqrQU#=RLi/H?8N5m2)\!oS#s6f>sQ!"MLb"E%ItE!uB',E%^1ruVZ%^R*q^QeKoNDN'1WMcAC/a%AAWC>Cm)!gi.cI9W8.&c!.ABf!T-9aCe4a#rr-J^DEQ&!dN^6$6,q]MsN/Aq)3MP7]^eR#ME^>?+&cf=hP\$o8m8n0B&HGR@n:GTQj;2pW>O%/%9F:GIKLR^3gs"0%`j$p6b2/[R2_lbo(9j2F*e4-kBCuYC7EPUOg)OmM6\KY1f^5Mtq&INg`>%(E4;U2D,FjtF8fF@+;*6h:r+f+C);($,hGa*("%>,#t@OF%s/mkrBdA6Gr,EM:eXk?"SOtr/#L=D*sSF)aL6Ed)dU2F"PnqAsf^sFD:Fm6pAp$I]NAEjhMrAA;"\O#/L2U3*]lIn^EM#K[$o_+5kTNr2eRI-FI6EesA_CY\UQ(5Kb_[[)lK"d-u_nSAiN"Zr1BCPC>Wl&.s^_tD._._94\X(#od18)qA%%#2.^MuJ(epT2e`*%/On]k$]aSn@k3#Q?)0Ar[s4hQMatA6*ll9csH(8c=W:]f@P^+)4qJ4O#J\t7BHia-qNSQ5cBL@u`s2?bBgI,)%#o/'rWY\BIuqN,],\X'>8gStS&ac:M99elaj,]1(cctkrKnICf_GUD2pe0Be^\Tcg*kpNfrf$3oGWU+1t`r->FV=Upq)GI?O%/n.c7E/=+/`s!XY,1HQ=5+L1q0mHIcrkdY%;:L]=i.jcMc/ZIr!oA8F5ggh]smIVK-:\VLd*p2,m,P-/%6SXnD;rf4mVikZ!Tj[LI\)qNbDT+L_T)Ibp_+38-/8m2%.HkoD4jM2(YfqeQ+5O5WI$;.1;Nc`&:lK>$K+V5IGbL"!]U`-+"2+^J-0f]JJ+:N3"Q6*-ug\d"&TV\&siYZ&/QE:.IRFa#C$],4TqcApO^!Z]En`RW9R[u/e7A+>\]q.XGL$;fa%UHiEcVlZ3#&XJ? /FormXob.cb203efccdde3cf25c323216122a748d 15 0 R /Length 7021 /ImageI ] /Type /Page >> 6 0 R /ImageI ] It is ( ) . /Length 3402 /Height 97 Doceri is free in the iTunes app store. endobj 23 0 obj 5 0 R % 'FormXob.de0bacdce49b01a092576fb42ea2758a': class PDFImageXObject Exponential functions occur frequently in physical sciences, so it can be very helpful to be able to integrate them. !30(8i@zzzzz!+=0'\ZH!o~>endstream Gb"0Td?<6,%#&-)s$LFFOdStqJ.Mp'_K8f:/DH3@eL7-)o?06IPXgtH7C+=M8GO.%:AU2,]h$cGJUNA3$s(XY+[\3O:ef,3NLE=lJLbR0"mH7^j(&/sf@`42bWC/-Dh'sI#MObiQfP]#$E7%V)u2+Jk5:X?&H)O/=FGIHX)7%"cieUgUDTACID,8"KlaXp]f[CZq#?^pJH!*/TKepL.8b.,fk7*a1#Ru$VXFAX.[?JZp5m5$A*AT9,_6Z:8S=[`'[6rpDuKf\X"42')0/0gUe[W5u7!qtaf3iD5;gJ5kKoGga.[FHcdUAV(!;-rU8QQrYI0:og;6]7Ec_FCE^\)8hla7hBNpptA^@gCG2mHVL;sopP]>Z'j),Zfa(?O,Bh>)V4>'g(Gc3u]No4ML6k?.=fshTRQD_-:iLp_f3eA;F6uRMJ(=?[QU:Q&Zd(7.k*W2uAX)oRiT$rRqc%7=JD:rnoI#-h7X5j18@;-r+J;AKL`AhL#p:dPS*l%F_e,W$_Q`$R@kSNNU-2EWk=_nL9g-/3V2@L*QnuT7;?]P`T(7]:2!q0/?`Z2oCVbd8)@sWm;MBk:qbU,@0,3G"`9$8?er#$Y'S[Vl;R`8d&dCIpdZPS[^TX>X/T7d6at8S?1e2P7FYg8Qo!NYR](b0IZ')AGdW7d=l3B/X/\'mGpRTkH?gg,,/FDDi)7*`p99b>_PCrSnhUZs#I+q9iTYI^'$`QBBf*ZlXK9O>H";]\N>\9"[#lonE_#+f^N(>(_DD)kbK4V3P&DCk*]S&36[7GrN_HUGrN1AXOK%6!O2LajP]rI\IKag9>_M3,j'D_!pX1>]j/M'gQc#s8SNl@Vcb?7haSrasXbC!?JnE][^a)Im8^+GV8L!\t+="I(Vfm5([^'%"[d^sA["Li+WYUjWI-!hj`=Ho:n-KurVc@9aq^m]9f$/4B.B=#[gR`ruuBM,t)'V2=WHEPY">`OJ00f8/"3ni*'),8)JYLk]M1)=R,mpC^:B">qtc]&-o.(N8q%(;S,$`WEIJ^eJrVn7hus,`dk9$No;rZN>[6mr2Z8!0H`S70! 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