7.2 Notes.pdf - l ”\“Q’V:2 W‘s Nomi 3(m a M tag...

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Unformatted text preview: l ”\“Q’V {:2 W‘s Nomi 3 (m a M tag, halls it Q at; K 7.2 — Trigonometric Integrals t Tri Identities You Must Kno / “J“ " WW“ "““'““""""””'“;m“°”’"‘i DANA [Ziwtfif‘s \lrrermYTqZZ—_Ti._/T,”A 1 w T srn e + cos 9 =1 srn29 = Ell—c0526) c0529 = §(l+cos?3lj i CA 3 b.0836” 1 a 3M 9) ‘3 \~ LogLQ’ » it but? 33 :3- \ wfigwf‘tzt 1% “Jr waders 29%,.” “Airgun Exam Ie :7 g A: ( \ + COS %X\(}x‘/\ : J” g \ “t“ (.335 <ijle l ~IC0524X dx : L w x3 r, LVT CT L Powers of Sines and Cosines - Odds These are of the form Isinpu cosqu du where at least one of the two numbers p and q is an odd positive integer. The other may be any real number. o If p is a positive odd integer, factor out sin u. Then write the rest of the integrand in terms of cos u by using the identitysinzu = l-COS2U. Then expand. The resulting integrals will all be of the form Ivndv where v = cos u and x dv = -sin u du. (9‘ If q is a positive odd integer, factor out cos u. Then write the rest of the integrand in terms of sin u by using the identitycoszu = l—sin2u. Then expand. The resulgng integrals will all be of the form Ivndv where v = sin u and . K \ ' dv=cosudu. q) ’1 MAX: 9091 Log X \ v s L Examples 3 \ LUS K Q\v 3 \r\ y l. Isrn xcos xdx :SE ”\(Q X MEX Q \T' Sal X\ (xx ‘ : Ssml’x msx (XX “' E 5 fix X W" ’L MK A: c5\,.\X (kw: L03 X 67a Page24 DUWQP (I? Slat). \3 0M» \NQ WI” UH WK :3ij “\1 L05 >4 3 WK 2. Isin5xcoszxdx :(L' 0?:y\1‘ L091 3g CJ‘W‘X “"‘ mu); “5&1 (05 K 3‘”\‘>4 (XX Ix: 93% {Q “A: UL QC”) CMZVSTTQU HR‘T — ‘\r =8 \i‘ in 5:45“? dd“ *3 ”(W ¥:'SK\IZ- LUIU‘ *lrxlfi’ik‘l :VKBg 233) :TUJFL’ Powers of Sines and Cosines— Evens These ore of The form Isinpu cosqu du where boTh p and q are non-negoTive even inTegers. In problems of This Type, we will use The following idenTiTies: sinze = %(l— c0529) and c0529 = —2]—( l+cos2e). m, . mmnwmwmc-mmwwlmm“ “MW“, ’l COS :‘l-l L05 “X. // USXTg‘ 3 ‘3 Inn-om rpm “WI van" By subsTiTuTing These Two idenTiTies and using olgebroic monipuloTions, we oTTempT To rewriTe The inTegrol in 0 form To which we con apply knowrl inTegroTi formulas. \NL \N\\\ “59 \&\§Lr\\\§\ Swlfi) {iii gig whéilupk W: FLU m5 5)ka ”“de 4.5;“ on i luau TIA. u: m Morino Page 25 WAX; Angle inaliliqé‘gl \ 3\r\ 2kz’lg \ruyblx) QUE—23¢ ”L’ZKH \(032.5{» Eight 2.]sioncosxdx :' 8‘1 R\ L031 ‘X\(\1\K\ SLOSZXX 3% \«l\ :: §Q\\Laslylk\ HWJXNM< WM 03:4 :Jakg K\ _, L03 2%) (pk Appw‘ 31 (A \t who‘ll Aifl (WWW \JSK M“ \M’AVH L05 ‘QZLQ (H noel?) WNW 9 1K “ —Lgo\xvlg Zm-Hasclxl (M \A L ¢K§\3§ “(MGM :glm lag-DEM X OS lfi‘ll” (win X {i vi MLUSMQB‘: 4 % <6 W) flak :imfi Vl3\r\\l\+ ln’te rals wi’rh Tan enl Colon enl Secan’t and Cosecan’r These inlegrols are attacked using the following fools: . Ilonxdx= Ifldx — -ln|cosxl+C cosx o J‘co’rxdx = I C.OSX dx = lnlsinx| + C Sinx . J'secz’xdx =lonx+C . Jcsczxdx = -co’rx+C . Isecx’ronxdx = secx + C . Icscxco’rxdx = -cscx + C Recall: sinze + c0526 :1 Divide by sinze: l+ c0129 = CSC29 Divide by c0529: lonQG +l = secze secx + lonx o Isecxdx = Isecx —-— secx + Tonx )dx = lnlsecx + lonxl + C cscx + co’rx . J-cscxdx = J‘cscx cscx + colx )dx = -lnlcscx + colxl + C Morino Page 26 mung—NV... Powers of TangenT, CoTangenT, SecanT and CosecanT - Evens These are of The form ITanpu secqu du where q is a posiTive even inTeger. We facTor ouTsec2U. Then wriTe The resT of The inTegrand in Terms of Tan u by using The idenTiTysec2u = T+Tan2u. If necessary, expand. The resuITing inTegraIs will all be of The form Iv”dv where v = Tan u and dv =seczu dU. Examples S‘QAZK +0021 akx : 840‘“? XRSLLZ'X‘V) ()V i. JTan xdx w _, EMA M m MW “MEX f» US: ATO\"\K 5‘“: SQL 7K6“ VQLK \ T _ M” .Yuv-wa—w u mun-M a‘.w..w ww—m,-mm~.Wn-WM, MW NW N...” ZJTanxsecxdxM$\\u¥ ‘YKL \3 (O‘TSM‘ 0 (Mn (Li/0M (DWTU) WL Wm ;6\L‘Hr* bad S‘chL‘ ighfix Sum Eu >4 (TX :<+0m\( “LXQTT‘M EA 394 :Eqknx m; x «M T 33mm >4 3% KW \LQ,‘\ \k: ATX\BL Oth— 39 XJVX Marina Page 27 Powers of Tangents and Secants — Odds These are of the form Itonpu secqu duwhere p is 0 positive odd integer and q is 0 positive integer. . Begin by toctoring out sec u ton u. Then write the rest of the integrand in terms of sec u by using the identityton2u = seczu-l. 0 You mag 8&0 find cotzu = csczu-l useful for some of these problems. Examples \\/ » . . l. J‘sec3xtonx\dx :1 g S QVQQVQS [L K, "'"i'li‘xfifil ) £34 m: Sic/)4 M: Sufi AMNX (ix , r' , 3 “Wmnmwummmxmum-«me “my.“ arm—1:. oil 2- liog’sxsecaxdx : 3 40.:qu Eulx K Arum 34 SM X (l K i 3 S KsmC'Sefi'UM/Qxél K‘th 3 39" M\ . :' Bfimé‘xvlseg 34 H Siéxj\ttox 3%“ W 93% WW swam ~1 X315 x Relax w; 330 “V X surly Gimme/weld Liz} w: SILL. >4 3%: 3 u. 5a.- "karma (Mk : 3W M a 1qu CM «ififi M. Merino Page 28 Powers of Tangents These are of the form Itonpu duwhere p is an integer and p > i. In problems of this type, we will repeatedly use the trigonometric identity: tonzu = secZU-l until we obtain either Itonu dU orIsec2u du. Example l. Iton5xdx :SAQAB K ‘td‘rg X CAVX :SAMXKSMV\ AM i. :Elrotn’sé ‘QLX AX V SAGA Xi!>4 \/\_~~~ ,‘AAAMX :S\(AHHKSQLXA\K \S Aonx“)i0\'\ X \AX — EARN? X 3% QXAXWSAANX\SMZ X W (AX M SAWQ X ECU. lXAX SAAoXSQszAX—kgiifi X AX L13" A :‘ ”WW AW?“ 5U, (W Morino Page 29 ...
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