Modul Integral Trigonometri

8/13/2019 Modul Integral Trigonometri

1/13

Rumus-rumus:

BENTUK DASAR

:.'.',,II. 5UB5TITUSLFUNGSI ATAU sUDUT.

iJsinxdx=-cosx+c Jcosxdx=sinx+::I Jsec'xdx=tanx+c Jcosec2xdx=i,r:, -:Jsecxtanxdx=secx+c Jcosec x cot x i, = -:::sa: .( - 3i Jtan x dx = lnlsec *l*. . -.r,r lcasx\ tc Jcot xdx=ln s,:'r :: * c . - r

Jsec x dx = lnlsec x + tan xl+ c Jcosec x dx = i- :csec x - cot xi + cJsin(ax + b)dx = -1.0r(ax + b)+ 6 fcos(ax + b t:, = -s::r(ax + b)+ cJ'aJsec2 (ax + b) dx = ltan (ax + b)+ c Jcosec'(ax+b)dx: (ax+b)+6Jsec(ax + b) tan(ax + b) dx = 1r..(u* + b) + c

J1U**r+cos2x)dx=iri-,3, ti5t,, ; ..i, == GZ' {.f(o.ffi 5sec 2x tan 2x)dx =i n\r( 4Y I ii" r'Y ' ;'-1 1 tt tc

3 4 I{*'(3--+)* tanls-2x)} ox =ls,n (i. x 4l - r, Ln tsec (v *tx)'l + t

fi J :rnix ,{x=6 J{.0..r 1s, + 5) - cot (4 - 5x)} ox =

trn lc.s.ctv*5)l - * tn \rrn (rt-sxi\ * c

1 Jsine 2x cos ?xdx = fi srr\,c:x*c 2 J.o.5 (3x * 5)sin(3x - 5)dx =-* rrsn (5r-i, 'r E3 Jsinz 5xcos5 5*d" =

J grnq 5v {I ..iti1i'i'' r:5 l, ;: 1ir.Jsfntsx qo:5x -r. llrt'\ lry. rr:s'*x "i linS lu co: r d'x| 1 i. rt' ltn'5x ,,, ',tr,' '.r .r

4 Jx cot ?x2 dx = ,o;*;,I x cot i, Ui, .), \r, lri,r 1r i - c ili "n .*ln ir,,,,,r'i , c

Created by Hartoyo, Integral lpA Page 16 of 31 7l24l2Ot2 I


2/13

" I secr/Y r,). [i 6 a "oi2u : rri, ;JJ -F-(]x -vx nr- .,)\i-{dx- ,t -i,-L lr1-Kdu,/ ln[s(crr iot(\+c\x: ln |sec, {i + tan tx. | *c

p cos ec(?x * 3) cot (2x - 3)J 7 +5cosec(2x_3) v -'.qese{:jlg _k, du*- :l r ] (.r - Caser. :_v- 5) re-f igr -j;l-t,':*,llr scosrc (rv-r)l* C

{ldr)iir

..Pc {jr:r--,^1;;1{:v fi,'lr rr.r l1 8

J 1o;to-;"^ :{ s6cr'}x rluL * a r* -l,,st )*^ r\'- ll1E t i.'ia;n; +Ci1 -(3: i to .( : z*) ' )' -J6cix= r" ".

J *'(r":a 9-j'j1'? ,tLi tu - -r - J (; lr. r (ir.,;*, fr" jl-lT(ti 1,,-ise


3/13

r25lnaxt5-; -ClX= dtpZsin? x+ 5 fr'4stnYcotr

f (- f)*,, srnx j0 &' 4q- ',:T: 4cor6td 4lnx coli l* du. *Jft' -* auf (] rnrertn,rtrl ' \(tsn"vtr)J 'tc (t/ t(r-- s z,) dxh t*- a on zx)+c'E^ - $r,n 1x t LIs,nxfi-,1-r*"*, *tlr Istn2x ax

r sin3 2x -.o"3 z*l----=--=----dx =J sint Zxcost 2xJton u geczx - cotl{ c.orec 2v dxb.s"o t* t b arcn Lx + c

Jsinssxax =Jstn s* (r-.osl56rSx - Eln 5{{srnsx -1f,,

J.ors 1+" -5)dx =Jcos (ax-si (l-etn"ux*)) dxIcos (ax-6)' cos tqx-s),n01 lcr-s) dxt srn (ax-s) - | ,,nt (4x-s) t c

Jsins 1z - 3x)dx =1 1'- s.f(z-3x)f srn 1z- 3r) olxf srn (z:3r)- 2 srn q2'4").cos1(tnx)r stv\(z-1il cass(z$ac a-bx) - \cose tt-30 * F cors (z-3")

J.os5 1s, + 3)dx =I (t- stnz (sx*31f cor (sr+ l) axI cos (sr+s) - rgtrP(sr+g) coslsx+g) + stnq[sxt$ cos (gr+:$ srn(sx*l) -b ,,nl (ex+31*|. srns 1sr+1) + 6.

Jsinnxdx otou Jcosnx{x positif gonjil, ubohloh menjodi pongkotdon

by Hartoyo, lntegral IPA Page 18 of 31 7l24l2l72,ii


4/13

Jsinz sxax =Jt(r-tos 6xJ dxtrx-?us,ng**sf.or2 o*d*J h t'n cor (oxl dxix+ fu srn lox + c

Jsina +xax =J t (,-co 6*)odxJi - l.@s Bx +{ str'' Ex dx,i^ - * rtn 8x + i I l('-*, lex)dx** -t stnE/t &* - k stnta x't C?*-*stnox-fur,n16r*c

f "osa f xax = "ry.,J h (r+cosfx)" dxtt. trorfqt cos'x &=.Yif *|cosir-th@pj;-

[.otzrG# z* -.ino z" a* =j-. r* stnzx fiffi;fx dx ' Ios'1t4xJi(rn *, z*) dxi*.f tto 2* +c

.os4 3x - "os6 3x dx =cos"bx{ll6Flx drJffio ar 41 . Jsrnrx (t-cosnsx)dxtrtn3x - stn3r cot2gx cl(

-cos 3:( t{-coss 9x i C

V. JtonnxdxJton3 l"a* =J tan tX ( ec'3x - l) dx,[tan 3x rectgx - ea,n lN dX[ *c"zx - tn lsec 3*\ + c

Jcot3sx ax ={61st (cosecaEx -r) #-icot rx Coseczsx - 4t 5rJgcoccagr -|tn \xn fx| * g

Jtonu (2 - 5x) dx =I ton ( z-rr) tsec"lr-* -lf dxI tnn lt-Sx1 -l se* 0-f 4tsn (r-6x) * sec4 Crcx).tsn{:-$x)*tn \ ,e. tz-sx)|t'| secai2-ExJ - fi secd(z-s4

Jcot5{+x + 3) dx =I cots (qr.e){cosec' (4x+g) - rl u*

.lcot? (qx +e). os ec' (4x*0 - cot (ztx'ts) fcosec" tax+ {- tl 1*j*tj h**r) co.n.z(46*3) ^ cot (ex* j)cusec{ (ev+:) 1 cot (4xtcota (axra)1f cot2(ax+3) t j 61lst.,xl + c

Gunokon rumus: 1.2.

sinzx = tF- cos2x)coszx = +(1+ cos2x)

; n bilongon positif gonjil, uboh menjodi pongkot

&eated by Hartoyo, lntegral IPA Page 19 of 31 7124120t2


5/13

Jtona zxax =Isect rx -l dxItonz*-x+C

Jcotz xdxJoseoz \x-Zue\x.- -r dxxtc

J(r*ton:x)z ax =I secl7x + 2to'tr 3r olv*+onty'Ih1ec3x\ +c

J(ton +*+sec 4x)2 dxI $ec1/v -l + 2sec4x tan zlxt sec? 4r d"xrztan ll x + lsec Arr - x3"Q*:'- _-a a.

J- -.r a- i-AA1LEEarF---U=".ra zrd, =I (,r+ tunozx) reco f.x clx\ sec'l* + 9ggtax -twta 2x dx)tanrx nt t0n3:.x *C

J"o"."a3x

Js".6 xdx:J (r+tan"x)' seclx drJ ser'x + ztqnax ,*11 + {rtn{{onx+?tonsxo? *i+x*

J.o ,".6 f x dx =fcore.o Ix ooseclf,x d/I (rcot"1x)'. cc,sec' x clrrI oosec'1>.-* ^t'f,x.coseclfx t.oka l-x -gJ/l?{hx dr


6/13

I [ -,-4 x,. ^^^2 d-. -t-. 4 f-,-5.- ---3 -- r-.JIf o,nt lox. sr.n1sx dxjtsnsex i(r*cos roX) dxf lrtn*lo{ - sln"tox cor

iox dxt J ('-*t zox')- rn2to x cos\qx dxt^ - f;o *,nroo- fu6 st,",1rox +C

IIf t,.P 2x srnex dxJt n,,. 2, z(t- cc - ,x; ci (Jt srt?rx - tr . i. -.r r-) t r cl,J t r*n 2x(r - cor r2a) - t ,,nt av cc: 2r' Jrf ft s,n2.x - t nn 1,( coraax - i. ltntl' 'gsc't< dr*.* 2xI )sscos32x- fi si"at*

i =:*-

jsinmxcosn xdx; m +n bilongon negotif bentuk ton x.I c Zdx [ 2secs(l_:,_ ,^ axJsin3x.os5x J tan2l -*i#'#- d-

2 J(r+ run'rn3tsn^x'toncx) ffi*eIua+ 3o-r* gu+ u3).re.'x S.*

U'tGnx49*. receld*. #..==%Vu-'td [n lul+ 3uA ' lr'-ro "c-cotzx + 6 trr l+cr,rx\o -qton'H%iLE. E

2#q

, Ue cpttd( ffi 'sos*axA* ; dP.-(osPf-'i1

t U,{f,xflVotu r. ^n(x'le*'

flu e* 3ub au-6d-L -SU , +c_6 o .r..-,F * f tonx \Gnx\lqnx tC

3dx 4 r 4dxt_=J sin5x.os3 xI A.Cosec9xI coe? y clxu. cdtxS,#sect1

4 cosecex (1* coe'x)o .1s---AiE x- -G*"rrlu-all)'t +30-2+ fi + 1 clu2tJ-'*120-t - tzln U - {g + c2tqnax + t2tqn 1 *'t? ln ltonxl -4 cot K +C

q pcr"i-l*d:

by Hartoyo, lntegral IPA Page 2l of 31 7l24l2gr2


7/13

1. sin s2. cos s3. cos q4. sin s

cos p = .}{sin(q*D) *sin(s-p)}sin p = l{sin(o*D)-sin(a-p)}cos P = r {9ot(q*p)*cos(s-p)}sin D = -+{cos(a*D)-cos(a-p)}

:. ;q_ , a ,?__, :...-*,, _.,e *- ,r,i.i,,: aX. Bentuk it4fuUef yqpg dopot disederhonokon',: ai"i 1. dengon mengolikon sekowonyo.

1 Jsin 3x cos 2x dx =f i(t'* 5x+ s{11/J d**.0, t'N* i'r':xtc

2 J + .or ?x sinTx dx =it{r,,., gx 'r Li.ti5s ) itr'1c':')'" '::" ;r

3 JS.in Z, sin 3x sin 4x dx =]'{ (.* 6x -. rn: 1* i.:*, : x'dx('* 5 ro:lx Str\.jf ,[ I :J 5 ce:6x lln jx * I ..

T *[.qe-r'gof k cos 3x dx ='t "?. "j] r - , " l.l\1.


8/13

3 t zdx tr-^^) 4 e 5 dx f q r^r:. 4 / --J l. r* 4" = J ' -ia,,r,r[- atJz *t'4x ccsgc 4x - 2 ri-:'.':' - 1. ' li clvJ rcot 4,< co$ec 4x .' acte '-', 4x -+ (x] cnre. z\x t cot 4x + ix +C

J 3.3"*".3": )tr' (.i13x? l*. 3x sec3x - tan? 3x dxt J,nn,a e(c'3Y - Fc'1x -+I cixi (+ t,.3x- ]3 rqp 3x tx ) t 6$ sec 3x* { *o,, e*.^ t *n.

xI

,1l

)a]J

I

II;II

IIIII

t_rCr

r. Bentuk Pecah ' dxtan Jfrvu"g daPat dn {r(x)} d* , r B {f '(x)} dxf(x) - J f(*)

eated by Hartoyo, lntegral IPA 1r -1 '-

bu

T3=

4

I,P

at daram".-":H*;-sa'i* B,*e- ' =

-:"*. -a 1.I r6cij*2x+?Bin2x, u:)si,r2{-.)ce))tll-:-OXL J iasin 2x =3 cos 2x (P . Arcl.lr r 6:ro-\FnOg;*r*r r -, r '1. ro' r,,11*ll*l'h*EE:.:j-.=--{.. ',: / -+--lA,,1iiffi.ie:r;'r'r J' {rs'.,.7".n1J.1dxI i-/ ou , - , ,A r .4(... .:-| ";t-" 1-s , 7[. ,". -,' ) \' /Ilf a Ar^, ,,,,6ttflJt r{,.'I j i :ix + j ?. "i:- ^- r'-'*---* 4e.l ro'i a s,,,r*l'liu+ t"1i, i2s'rr ,( -3co: lx\ * cI

I

II

II

III r 36cos2x - 2sin2x .tl: 3cts7x*ttstt\2*lJ a x o. 3 g:-(5rn2*'r6s"'2{I iA(3(errxr4rr,' ;x) iL(-(..,r,iri d"''-tr iI J tCArry +4:,,1 1x dv )ntc-;* rq',r.....I fl uAU" ,g Ah -',b='z l=l e**ret 1L (t -gY"J 15'5II Ja crx "/g t 0,.;I| 4x t 3\n\ lrl :x r 41r;.;i1*(I

I

II

III

age 23 of 31 7l24l2ot2


9/13

J {rin z* - 2 cos 3x)2 dxJsrn'zr -4os9x. sto2I +4cosa3x dxJ | (s - cos ax) - 2( srn 5x- stn x) n 4 f, .(r*co,: ar\ d"Ji- i.or 4x - 2trnlxt astnx t l* L.rrt a* dt*^ - a.ot * t f, sto 4x +Z cosrx *N srrr[xtc

Jtrin 2x-cosec 'x)2 dxI t*bx -4.0. X +osq.26J dxJt-lo.4x -Acosx tcosqc2x dx|y - |srnAx - 4srnK- cotx +C

Jta.or 4x-secx)2 dxgr*'qrt seclX -Lqszlx 'secx dxI n .r8x + sec2x-6cosaJr 5scx16stn21r secxfJ} * l^, 8x + soor x - 6 (os"x- stn2x)r rcx t 24 srrPx cosx &I1 . $ * 8x + teca x - G eec N + lz stnlx.cosx + 24 $ nax cos xJa- 1* 31 treoz{ -6 rcx t36 s(nax sx dx\| . fustndx t ahx -c tn lsecx t te"rt\ t llsto3x t

Jtun * tan 2x tan 3x dx1o61r t0.h2r( .,

-' /) 't'ahxi'tor\ 3x- tqn 3x +ctn 2rton3i- ttrn 2x - tonx.tonYI ton g ,c - ton 2x - tenx dxthlsecg*l- {tn1sec

r1-cos 4xldxJ 1+cos 4xJ ffi$Ha' icuoqzx d^J se61 zx- t drb,+oor1.;* -x tc

J@JCosZn+stn2x dx :*r,n 1x-, c,,, Lx + c

Jsin 2x cos x cos 2x cos-4x cos px dxI corr 5\N \6/ dx/'r(o" r1x* rlh tsx) ilxh ,0, tlx - lo cos lrx +c

by Hartoyo, lntegral IPA Page 24 of 31 7l24lm1'


10/13

f 1 ^,^r Att -- - U ' :ttt \|zx 10 I Sin 2x r.- - .fl: tff}x-i./r-"r YE^ vvr Ya UA :

- t,l^.,\= '1"i.V (lr-=' \' ('li \1y :-',.r 'i.). /':'-. r -- \:/

-/^^. l:,- - tl J 11.

; :a, C i 1, a ffi t(

J sin (x + f) sin(x - f,)I srn:vI-

a"i(t,- iY-{' "\ ',at

dr,'i1''r'-\\',')v

.ii. ar, trl Ur t' ,,, 1

ftun*sinxtanS dx. ':1., i,(rt{ r*: :- , A,

"fIA

tli,':,t1 i9'u:ls :j(1{qrty.+nr\itcr r' ioi, .r* . tn{


11/13

L7 r sin x+sin 2x 1B t,cos x+sin x cos x-sin x.J 1+cos x+cos 2x[ 51nx.t 1t , Y (-J Y ( Srr,v (1r :; r,:. ;J ;';; *?ccn"- r.- o\x',l .;;;,....,; ''"Itnn n ild . {r, lsrc xltc

J'cosx-sinx cosx-snx| 4rj',{J$-'. ^, . . '--:--::-J .'r1l /- gtne / (/ Y t c:- ''ln lrec l,rl +' c

19 r sin3 x + cos3 xl-=--- dxJ SINX + COSr (5'nx. lce:x;(;'r''x-:{rrv (i'xt((:1'I l\,*,.J l,,ti7 {f :'

rJl'1.,i-i ;r

1{ ,,{,..A 'rv ' A*Ltr/ ' E.

,r" ?:.-*.1.{ 1.

20.l v-rIr

Lotidon ir

*l Integrol Trigonometri 1rokon oensil. tipex don kolkulotorer i sr

1 J(0.o.".10x+3secI [n [^,r. lcrx -cot t0x tr'g ? cosz (4 - 3x)I \ / ,{v-I {sin(8 - 6x)}10 - 4cosec(3x - 4)r cti('ix-A) dx u' to-aco.qtli' ttt* a* ,1, ,..-.J-e,,n(Ar


12/13

l .^o5 lu^ ,1- - 6 11) oin { o;- X nnol -.1--i*,.o.ffi, dx = Jcor'2a srn'2,x dKLroc*')Y +C a (ocsf - er+)o, ia"J-g (*r* *^ ?) + 3(lror i)o.=Zsvt X - $ sn T *j* + gsrr$ +C

j

7 gaJ ilra+".*'+,r A secqAx dxl-1.6 =r\f uns axl,]J#q"+ .#-, Jui *ut du

I

3qKelos/No: I ^.* 5*e l Nd?na: S Niloi:

LotihqruAUiuroggKerjakon denson sinqkot don ieiFqtohhirerfS Integrol Trigonometri 2rkon pensil, tipex don kolkulotor1 n,i u'tqn 4Y*;*r'qntt 2 Jcot6 sx dx = f (csec , k. -l? dx

"[bsco'ax (trot'ax)hxJcoseolar t zcob*bx case;,bx+6p4y.cpset.3x dd-trtt* -$ adz*-#*t bx +c

3 Jrors aro, = J ,os ax (t- tn'ex)o 4r.I *, un - 2$n23( cos 3x + $'nqex at 1x dxIs.nax-f smtTy+ f srntex+c

4 J.osu z* x=Jroutrr{ffi* ,,Jr*r:* stn 2t 4r1tr*t2x+G

Icreated by Hartoyo, lntegral IPA page 27 of3t Tlz4lzorz


13/13

I ^^^+ q-.r^-7 q.. s,. (, I ltn 4^^^^ rv *,J\ serru(r*ton"g) +on? lx dxI setx+qnf z* +secl2rtn9 2xia ,on u rx * t tqn'2x +c'

fi )'tr ftbtdrr{ dxi\E J stn 2x dx\fF cos ,.x + c

7 I*-on,-o'= Jlsn"'e*6a* *n 4x +e

I

=q

%= ={hflx-:1q 4x axie. ',=Fru'5ot Ax- @:ax srrra4x dxfi srrf 4x +c

9 t- x x 1 .'"iI U stn - stn - cos-x dx =J 2 3 6jIt@'T i6:Eb1 xI^zutx-t ?a*-2srnX -3Stn + 6fitrb *C

Coret-coretan:

Kelos/No: Nomo: Niloi:

Created by Hartoyo, tntegral IPA Page 28 of31 7ll4l2Vt2

Modul Integral Trigonometri

Documents

Transcript of Modul Integral Trigonometri