Kef laio 1 To Aìristo Olokl rwma - eap.gredu.eap.gr/pli/pli12/shmeiwseis/Oloklhrwmata_2.pdf2 KEF ALAIO…

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  • Keflaio 1

    To Aristo Oloklrwma

    1.1 Orismc

    Ac upojsoume ti mac ddetai ma sunrthsh f(x), gi thn opoa gnwrzou-me ti qei prokyei san pargwgoc mic llhc sunrthshc (x), dhladd(x)

    dx= f(x) ka ti h f(x) enai gnwst kai h (x) gnwsth. Zhtetai na

    prosdiorsoume thn (x), dojeshc thc f(x). 'Ena ttoio prblhma lgetaidiaforik exswsh . H lsh thc enai mi llh sunrthsh, po onomzetaiaristo oloklrwma kai sumbolzetai

    f(x)dx. Genik:

    Jerhma 1.1 Kje sunrthsh f(x), suneqc sto anoikt disthma (a, b),enai oloklhrsimh sto (a, b).

    Jerhma 1.2 Kje sunrthsh, qousa aristo oloklrwma sto (a, b), jaqei peira arista oloklhrmata, ta opoa ja diafroun metax touc katma stajer posthta C.

    Epomnwc, o prohgomenoc tpoc genikeetai wc exc:

    d(x)

    dx= f(x) (x) =

    f(x)dx + C

    1.3 H Stajer thc Oloklrwshc

    H stajer C, pou anaframe prohgoumnwc apotele domik kai anapspastostoiqeo tou oloklhrmatoc kai kal enai na mhn paralepetai pot. 'Otanddontai orismnec plhroforec (arqikc sunjkec), h stajer aut mporena prosdiorisje.

    1

  • 2 KEFALAIO 1. TO AORISTO OLOKLHRWMA

    1.4 Stoiqeidh Arista Oloklhrmata

    Qrhsimopointac ton orism, mporome na apodexoume touc paraktw t-pouc twn stoiqeiwdn aorstwn oloklhrwmtwn. Prgmati, paraggish twndeutrwn meln mac ddei pnta ta prta.

    xadx =

    xa+1

    a + 1+ C, a R, a 6= 1 ,

    1xdx = ln |x|+ C

    exdx = ex + C

    xdx = x + C ,

    xdx = x + C

    dx2x

    = x + C , dx

    2x= x + C

    dx1 x2

    = ox + C , dx

    1 x2= ox + C

    dx1 + x2

    = ox + C , dx

    1 + x2= ox + C

    1.5 Basikc Idithtec

    Isqoun oi paraktw idithtec:

    [f(x) + g(x)]dx =

    f(x)dx +

    g(x)dx

    f(x)dx =

    f(x)dx, R

    'Opwc blpoume den uprqei idithta pou na perigrfei thn sumperifor touoloklhrmatoc sqetik me to ginmeno sunartsewn. Aut enai kai h aitathc duskolac upologismo oloklhrwmtwn, en antijsei me thn paraggishpou akolouje sugkekrimno algrijmo.

  • 1.6. LUMENES ASKHSEIS. 3

    1.6 Lumnec Askseic.

    1.1 Upologsate to aristo oloklrwma:(4x3 5x2 + 6x 1)dx.

    Lsh: Diadoqik qoume,(4x3 5x2 + 6x 1)dx =

    4x3dx

    5x2dx +

    6xdx

    1dx =

    = 4

    x3dx5

    x2dx+6

    xdx1

    dx = 4x3+1

    3 + 15 x

    2+1

    2 + 1+6

    x1+1

    1 + 11 x

    0+1

    0 + 1+C =

    = 4x4

    4 5x

    3

    3+ 6

    x2

    2 1x

    1

    1+ C = x4 5

    3x3 + 3x2 x + C

    q.e.d.

    1.2 Upologsate to aristo oloklrwma:(

    x + 13x)dx.

    Lsh: Diadoqik qoume,(

    x +13

    x)dx =

    x

    12 dx +

    x

    13 dx =

    =x

    12+1

    12

    + 1+

    x13+1

    13

    + 1+ C =

    2x

    x

    3+

    3

    23

    x2 + C

    q.e.d.

    1.3 Prosdiorsate kamplh, dierqomnh ap to shmeo M(1, 5) kaim klsh 4x.

    Lsh: 'Estw y = f(x) h exswsh thc sugkekrimnhc kamplhc. Afo h klshthc enai 4x ja qoume df

    dx= 4x. Me oloklrwsh qoume f(x) =

    4xdx =

    2x2+C. Epeid to shmeo M ankei sthn kamplh, petai ti f(1) = 5, opte5 = 2 + C, C = 3. 'Ara telik, h zhtoumnh kamplh enai h f(x) = 2x2 + 3.

    q.e.d.

    1.4 Ta oriak soda mac epiqerhshc ddontai ap thn sqsh:

    dR

    dq= 22 4q + 7q

    En thn qronik stigm mhdn soda den uprqoun, brete thn su-nrthsh esdwn.

  • 4 KEFALAIO 1. TO AORISTO OLOKLHRWMA

    Lsh: Ap thn sqsh dRdq

    = 22 4q + 7q qoume ti

    R(q) =

    (22 4q + 7q)dq

    R(q) = 22q 4 q2

    2+ 7 q

    3/2

    3/2+ C = 22q 2q2 + 14

    3 q3/2 + C

    Mac ddetai akma ti R(0) = 0, ap pou prokptei ti C = 0 kai telik:

    R(q) = 22q 2q2 + 143 q3/2

    q.e.d.

    1.7 Askseic Proc Eplush.

    Upologsate ta paraktw oloklhrmata:

    1.5 4x35x2

    x2dx.

    Ap: 2x2 5x + C.

    1.6(2x2 1

    x2+ x)dx

    Ap: 23x3 + 1

    x+ x

    2

    2+ C.

    1.7(4

    x 5x2 + 6x 12 1)dx

    Ap: 83x

    32 5

    3x3 + 12

    x x + C.

    1.8( 1

    x4+ 14x 4)dx

    Ap: 13x3

    + 43

    4

    x3 4x + C.

    1.9( 3

    x xx

    + 7x2

    3x2

    x)dx

    Ap: 34x

    43 2

    3x

    32 + 3x

    73 2

    3x

    32 + C.

    1.10(ex xe)dx

    Ap: ex xe+1e+1

    + C.

    1.11 Prosdiorsate kamplh, dierqomnh ap ta shmea (1, 2), (14,10) ka

    m klsh antistrfwc anlogh tou x2.

  • 1.7. ASKHSEIS PROS EPILUSH. 5

    Ap: f(x) = 4x

    + 6.

    1.12 En h sunrthsh oriako kstouc enai 5 6q2 + 7q3, brete thn su-nrthsh tou kstouc.

    Ap: 5q 2q3 + 74q4 + C

    1.13 En h sunrthsh oriakn esdwn enai 16 4q 1q, brete thn sunr-

    thsh twn mswn esdwn.

    Ap: 1q(16q 2q2 ln q + C)

    1.14 H oriak zthsh enc prontoc ddetai ap thn sqsh: dDdp

    = 0.1.En gnwrzoume ti, tan h tim tou prontoc enai 3 qrhmatikc mondec,zhtontai 11 mondec prontoc, brete ta olik soda tan h tim ja enai 4.5qrhmatikc mondec.

    Ap: 11.3p 0.1p2

  • 6 KEFALAIO 1. TO AORISTO OLOKLHRWMA

  • Keflaio 2

    Oloklrwsh me thn Mjodo twnProsdioristwn Suntelestn

    2.1 H Mjodoc

    H mjodoc aut efarmzetai sunjwc se oloklhrmata thc morfc:

    I =

    (x)(x)dx

    pou (x) polunumo kai (x) ekjetik sunrthsh. (p.q. ex, 2x, 5x2.) Jew-

    rome ti to oloklrwma pou zhtme, I, grfetai:

    I = (Axk + Bxk1 + xk2 + + E) (x)

    pou k nac, katllhla epilegmnoc, bajmc poluwnmou. Afo to I enaiaristo oloklrwma, ja isqei profanc h sqsh:

    dI

    dx= (x)(x)

    Antikajistntac to I me to so tou kai knontac prxeic, prosdiorzoumetouc suntelestc A, B, , . . . kai telik to I.

    2.2 Lumnec Askseic.

    2.1 Upologsate me thn mjodo twn prosdioristwn suntelestnto oloklrwma:

    x3e2xdx.

    7

  • 8KEFALAIO 2. OLOKLHRWSHME THNMEJODOTWNPROSDIORISTEWN SUNTELESTWN

    Lsh: Upojtoume ti to zhtomeno oloklrwma I, qei thn morf:

    I = (Ax3 + Bx2 + x + )e2x

    pou A, B, kai suntelestc proc prosdiorism. Ap thn sqsh: dIdx

    =x3e2x, qoume:

    (Ax3 + Bx2 + x + )e2x + (Ax3 + Bx2 + x + )(e2x) = x3e2x

    ra,

    (3Ax2 + 2Bx + )e2x + 2(Ax3 + Bx2 + x + )e2x = x3e2x

    (2A)x3e2x + (3A + 2B)x2e2x + (2B + 2)xe2x + ( + 2)e2x = x3e2x

    Exisnontac touc suntelestc twn do mern thc exswshc, parnoume tossthma:

    2A = 1

    3A + 2B = 0

    2B + 2 = 0

    + 2 = 0

    ap' pou brskoume:

    A =1

    2, B = 3

    4, =

    3

    4, = 3

    8

    kai ra to zhtomeno oloklrwma enai:

    I = (1

    2x3 3

    4x2 +

    3

    4x 3

    8)e2x + C

    q.e.d.

    2.2 Me th mjodo twn prosdioristwn suntelestn upologsateto oloklrwma: I =

    5x22xdx.

    Lsh: Jewrome ti: I = (Ax2 + Bx + )2x. Ap thn sqsh dIdx

    = 5x22x,qoume:

    (Ax2 + Bx + )2x + (Ax2 + Bx + )(2x) = 5x22x

    (2Ax + B)2x + (Ax2 + Bx + )2x ln 2 = 5x22x

  • 2.2. LUMENES ASKHSEIS. 9

    Exisnontac touc suntelestc parnoume:

    A ln 2 = 5

    2A + B ln 2 = 0

    B + ln 2 = 0

    ap' pou prokptei ti:

    A =5

    ln 2, B = 10

    (ln 2)2, =

    10

    (ln 2)3

    kai ra to zhtomeno oloklrwma enai:

    I =

    (5

    ln 2x2 10

    (ln 2)2x +

    10

    (ln 2)3

    )2x + C

    q.e.d.

    2.3 Upologsate to oloklrwma I =

    x2x2dx.

    Lsh: Jewrome ti to zhtomeno oloklrwma I, qei thn morf: I =(Ax + B)x

    2. H sqsh dI

    dx= x2x

    2, ja dsei:

    (Ax + B)x2

    + (Ax + B)(x2

    ) = x2x2

    opteAx

    2

    + (Ax + B)2xx2

    ln = x2x2

    Exisnontac touc suntelestc qoume:

    2A ln = 1

    A + 2B ln = 0

    ap' pou brskoume ti:

    A =1

    2 ln , B = 1

    4(ln )2

    ra to zhtomeno oloklrwma enai:

    I =

    (1

    2 ln x 1

    4(ln )2

    )x

    2

    + C

    q.e.d.

  • 10KEFALAIO 2. OLOKLHRWSHME THNMEJODOTWNPROSDIORISTEWN SUNTELESTWN

    2.4 Upologsate to oloklrwma: I =(x + 1)1000(5x + 7)dx.

    Lsh: Upojtoume ti to I qei thn morf I = (x + 1)1001(Ax + B). Hsqsh dI

    dx= (5x + 7)(x + 1)1000 ja mac dsei:

    (Ax + B)(x + 1)1001 + (Ax + B)((x + 1)1001) = (5x + 7)(x + 1)1000

    A(x + 1)1001 + (Ax + B)1001(x + 1)1000 = (5x + 7)(x + 1)1000

    A(x + 1)(x + 1)1000 + (Ax + B)1001(x + 1)1000 = (5x + 7)(x + 1)1000

    kai aplopointac to (x + 1)1000, parnoume:

    Ax + A + 1001Ax + 1001B = 5x + 7

    ap' pou exisnontac touc suntelestc, prokptei:

    A =5

    1002, B =

    7009

    1003002

    'Ara, to zhtomeno oloklrwma enai:

    I =(

    5

    1002x +

    7009

    1003002

    )(x + 1)1001 + C

    q.e.d.

    2.3 Askseic Proc Eplush

    2.5 Upologsate to oloklrwma

    6x6exdx.

    Ap: (4320 4320x + 2160x2 720x3 + 180x4 36x5 + 6x6)ex + C

    2.6 Upologsate to oloklrwma

    11x2e2x+7dx.

    Ap: e2x+7(114 11

    2x + 11

    2x2) + C

    2.7 Upologsate to oloklrwma

    6x3xdx.

    Ap: 6x

    ln a(3

    2+ 2x 3

    2x2 + x3) + C

    2.8 Upologsate to oloklrwma

    x25x3dx.

    Ap: 13 ln 5

    5x3

    2.9 Upologsate to oloklrwma(2x + 8)999(7x + 9)dx.

    Ap:(

    72002

    x + 89622002000

    )(2x + 8)1000

  • Keflaio 3

    Oloklrwsh me Antikatstash

    H oloklrwsh me antikatstash baszetai kurwc ston paraktw metasqh-matism:

    f(g(x))g(x)dx =

    f()d oo = g(x)

    Paratrhsh 3.1 H oloklrwsh me antikatstash efarmzetai sunjwc sesunartseic pou na tmma touc enai snjeto, thc morfc f(g(x)) kai to lloperiqei thn pargwgo thc g(x), mporome ekola na thn sqhmatsoume.

    3.1 Lumnec Askseic

    3.1 Upologsate to aristo oloklrwma: dx

    2x+3.

    Lsh: Jtoume = 2x + 3, opte ddx

    = 2, d = 2dx kai to oloklrwmagnetai

    dx2x+3

    = d

    2= 1

    2

    d

    = 12ln ||+ C = 1

    2ln |2x + 3|+ C.

    q.e.d.

    3.2 Upologsate to aristo oloklrwma: dx

    45x .

    Lsh: Jtoume = 4 5x, opte ddx

    = 5, dx = d5 kai to oloklrwmagnetai

    dx45x =

    d5 =

    15

    d

    = 15ln ||+ C = 1

    5ln |4 5x|+ C.

    q.e.d.

    3.3 Upologsate to aristo oloklrwma: dx

    (32x)3 .

    11

  • 12 KEFALAIO 3. OLOKLHRWSH ME ANTIKATASTASH

    Lsh: Jtoume = 3 2x, opte ddx

    = 2, dx = d2 kai to oloklrwmagnetai

    dx(32x)3 =

    d23 =

    12

    d3

    = 12 3+13+1 + C =

    12 122 + C

    = 14(32x)2 + C.

    q.e.d.

    3.4 Upologsate to aristo oloklrwma: xdx

    x2+1.

    Lsh: Jtoume = x2 + 1 kai qoume ddx

    = 2x, opte dx = d2x

    kai tooloklrwma diadoqik gnetai:

    xdxx2 + 1

    = xd

    2x

    =

    1

    2

    d

    =1

    2ln ||+ C = 1

    2ln(x2 + 1) + C

    q.e.d.

    3.5 Upologsate to aristo oloklrwma:(1 2x)100dx.

    Lsh: Jtoume = 1 2x kai qoume ddx

    = 2, opte dx = d2 kai tooloklrwma diadoqik gnetai:

    (1 2x)100dx =

    100

    d

    2= 1

    2

    101

    101+ C = 1

    202(1 2x)101 + C

    q.e.d.

    3.6 Upologsate to aristo oloklrwma:

    ex3+2x2dx.

    Lsh: Jtoume = x3 + 2, opte ddx

    = 3x2, dx = d3x2

    kai to oloklrwmadiadoqik gnetai:

    ex3+2x2dx =

    ex2

    d

    3x2=

    1

    3

    ed =

    1

    3e + C =

    1

    3ex

    3+2 + C

    q.e.d.

    3.7 Upologsate to aristo oloklrwma:

    3xdx.

    Lsh: Jtontac = 3x, qoume ddx

    = 3x ln 3, dx = d3x ln 3

    kai to oloklrwmadiadoqik gnetai:

    3xdx =

    3xd

    3x ln 3=

    1

    ln 3

    d =

    1

    ln 3 + C =

    1

    ln 33x + C

    q.e.d.

  • 3.1. LUMENES ASKHSEIS 13

    3.8 Upologsate to aristo oloklrwma:

    xdx.

    Lsh: Kat' arqc grfoume x = xx

    kai met jtoume = x, opteddx

    = x, dx = dx

    kai to oloklrwma diadoqik gnetai:

    xdx =

    xx

    dx = x

    x dx

    = d

    = ln ||+C = ln |x|+C

    q.e.d.

    3.9 Upologsate to aristo oloklrwma: 1

    2xdx.

    Lsh: Jtontac = x, qoume ddx

    = 12x

    , dx = 2xd kai to olo-klrwma diadoqik gnetai:

    12x

    dx = 2xd

    2x=

    d = + C = x + C

    q.e.d.

    3.10 Upologsate to aristo oloklrwma: 3xdx

    4x.

    Lsh: Jtontac = x, qoume ddx

    = x, dx = dx

    kai to oloklrwmadiadoqik gnetai:

    3xdx4x

    = 2xxdx

    4x= (1 2x)xdx

    4x=

    = (1 2)x d

    x

    4= d

    4 d

    2= 1

    33+

    1

    +C = 1

    33x+

    1

    x+C

    q.e.d.

    3.11 Upologsate to aristo oloklrwma: xxdx

    xx+x.

    Lsh: Jtontac = x x+x, qoume ddx

    = x x, x xdx = dkai to oloklrwma diadoqik gnetai: xxdx

    xx + x= d

    = ln ||+ C = ln |xx + x|+ C

    q.e.d.

    3.12 Upologsate to aristo oloklrwma: dx

    x(ln x+3).

  • 14 KEFALAIO 3. OLOKLHRWSH ME ANTIKATASTASH

    Lsh: Jtontac = ln x + 3, qoume ddx

    = 1x, d = dx

    xkai to oloklrwma

    diadoqik gnetai: dxx(ln x + 3)

    = d

    = ln ||+ C = ln | ln x + 3|+ C

    q.e.d.

    3.13 Upologsate to aristo oloklrwma: x2dx

    4x3+2 .

    Lsh: Jtontac = x3 + 2, qoume ddx

    = 3x2, d = 3x2dx kai to olokl-rwma diadoqik gnetai:

    x2dx4

    x3 + 2=

    1

    3

    d4

    =

    1

    3

    34(

    34

    ) + C = +49(x3 + 2)

    34 + C

    q.e.d.

    3.14 Upologsate to aristo oloklrwma:

    x2 2x4dx me 0 0.

    5.34 Dexate ti, en to p(x) enai polunumo n-bjmo, tte:exp(x) = ex[p(x) p(x) + p(x) + (1)np(n)(x)]

  • 34 KEFALAIO 5. PARAGONTIKH OLOKLHRWSH

  • Keflaio 6

    Anagwgiko Tpoi

    6.1 Genik

    Me thn bojeia thc paragontikc oloklrwshc mporome na apodexoumetpouc anagwgc enc oloklhrmatoc se llo aplostero. Auto oi tpoionomzontai anagwgiko.

    6.2 Trigwnometriko Anagwgiko Tpoi

    Oi ktwji tpoi enai qrhsimtatoi:En I =

    xdx, tte:

    I = 1xx

    +

    1

    I2

    En I =

    xdx, tte:

    I =1xx

    +

    1

    I2

    6.3 Anagwgikc Tpoc Rhtc Oloklrwshc

    O ktwji tpoc enai exairetikc spoudaithtoc, afo qrhsimopoietai stonupologism oloklhrmatoc, rhtc sunartsewc.'Estw I =

    dx(x2+a2)

    , tte:

    I =1

    a2

    [x

    (2 2)(x2 + a2)1+

    2 32 2

    I1

    ]

    tan diforo tou 1.

    35

  • 36 KEFALAIO 6. ANAGWGIKOI TUPOI

    6.4 Lumnec Askseic.

    6.1 Upologsate to

    5xdx.

    Lsh: Ja qrhsimopoisoume ton sqetik anagwgik tpo. Jtoume I5 =5xdx kai qoume, antikajistntac ston I =

    1xx

    + 1

    I2:

    I5 =4xx

    5+

    5 15

    I3

    Gia to I3 qoume:

    I3 =2xx

    3+

    3 13

    I1

    kaiI1 =

    xdx = x + K

    Me antstrofh antikatstash kai ektlesh twn prxewn, qoume telik:

    I5 =4xx

    5+

    42xx

    15+

    8

    15x + C

    q.e.d.

    6.2 Upologsate to 1

    (x2+3)3dx .

    Lsh: ja qrhsimopoisoume ton anagwgik tpo rhtc oloklrwshc.'Estw I3 =

    1(x2+3)3

    dx tte:

    I3 = 1

    (x2 + (

    3)2)3dx

    , opte =

    3 kai diadoqik qoume:

    I3 =1

    (

    3)2

    [x

    (2 3 2)[x2 + (

    3)2]31+

    2 3 32 3 2

    I2]

    I2 =1

    (

    3)2

    [x

    (2 2 2)[x2 + (

    3)2]21+

    2 2 32 2 2

    I1]

    I1 = 1

    x2 + (

    3)2dx =

    13o

    x3

    + K

    Me antstrofh antikatstash kai ektlesh twn prxewn, qoume telik:

    I3 =x

    12(x2 + 3)2+

    x

    24(x2 + 3)+

    1

    24

    3o

    x3

    + C

    q.e.d.

  • 6.4. LUMENES ASKHSEIS. 37

    6.3 Upologsate to 1(x2 + x + 2)2

    dx

    Lsh: Ja gryoume kat' arqc to x2 + x + 2 san jroisma tetragnwn:

    x2 + x + 2 = x2 + 21

    2x + 2 = x2 + 2

    1

    2x +

    1

    4 1

    4+ 2 =

    =(x +

    1

    2

    )2+

    7

    4=(x +

    1

    2

    )2+

    (7

    2

    )2

    To oloklrwma tra grfetai:

    I2 = 1[(

    x + 12

    )2+(

    72

    )2]2dx

    Qrhsimopointac ton anagwgik tpo qoume:

    I2 =1(7

    2

    )2 x +

    12

    (2 2 2)[(

    x + 12

    )2+(

    72

    )2]21 + 2 2 32 2 2 I1

    I1 = 1[(

    x + 12

    )2+(

    72

    )2]dx = 27o(

    2x + 17

    )+ K

    Me antstrofh antikatstash kai ektlesh twn prxewn, qoume telik:

    I2 =2x + 1

    7(x2 + x + 2)+

    4

    7

    7o

    (2x + 1

    7

    )+ C

    q.e.d.

    6.4 Upologsate to

    1(5x2 + x + 1)3

    dx

  • 38 KEFALAIO 6. ANAGWGIKOI TUPOI

    Lsh: To oloklrwma diadoqik grfetai: 1(5x2 + x + 1)3

    dx = 1

    53(x2 + 1

    5x + 1

    5

    )3dx =

    =1

    125

    1(x2 + 1

    5x + 1

    5

    )3dxMetatrpoume to x2 + 1

    5x + 1

    5se jroisma tetragnwn :

    x2 +1

    5x +

    1

    5= x2 + 2 1

    2 15x +

    (1

    10

    )2(

    1

    10

    )2+

    1

    5=

    =(x +

    1

    10

    )2+

    19

    100=(x +

    1

    10

    )2+

    (19

    10

    )2To oloklrwma tra grfetai:

    I3 =1

    125

    1[(x + 1

    10

    )2+(

    1910

    )2]3dxO anagwgikc tpoc ja dsei telik:

    I3 =10x + 1

    38(5x2 + x + 1)2+

    15(10x + 1)

    361(5x2 + x + 1)+

    300

    361

    19o(

    10x + 119

    ) + C

    q.e.d.

    6.5 En I =

    xdx, dexate ti

    I = 1xx

    +

    1

    I2

    Lsh: Ergazmenoi paragontik qoume:

    I =

    1xxdx =

    1x(x)dx =

    = 1xx

    [x(1x)]dx =

    = 1xx +

    [x( 1)2xx]dx =

    = 1xx + ( 1)

    [2x2x]dx =

  • 6.4. LUMENES ASKHSEIS. 39

    = 1xx + ( 1)

    [(1 2x)2x]dx =

    = 1xx + ( 1)

    [2x]dx ( 1)

    [x]dx =

    = 1xx + ( 1)I2 ( 1)I

    Lnontac autn thn teleutaa exswsh wc prc I , parnoume ton zhtomenoanagwgik tpo.

    q.e.d.

    6.6 En I =(x2 + a2)dx, dexate ti

    I =x(x2 + a2)

    (2 + 1)+

    (2a2)

    (2 + 1)I1

    , 6= 12.

    Lsh: Oloklhrnontac paragontik, qoume diadoqik:

    I =

    [x(x2 + a2) ]dx =

    = x(x2 + a2)

    x[(x2 + a2) ]dx =

    = x(x2 + a2)

    x(x2 + a2)12xdx

    = x(x2 + a2) 2

    (x2 + a2)1x2dx

    = x(x2 + a2) 2

    (x2 + a2)1(x2 + a2 a2)dx

    = x(x2 + a2) 2

    (x2 + a2)dx + 2a2

    (x2 + a2)1dx

    = x(x2 + a2) 2I + 2a2I1

    Epilontac thn telik sqsh, c prc I , parnoume to zhtomeno.

    q.e.d.

    6.7 Brete anagwgik tpo gia to oloklrwma(ln x)dx.

  • 40 KEFALAIO 6. ANAGWGIKOI TUPOI

    Lsh: 'Estw I =(ln x)dx. Diadoqik qoume:

    I =

    x(ln x)dx =

    = x(ln x)

    (ln x)1x1

    xdx =

    = x(ln x)

    (ln x)1dx = x(ln x) I1

    o opooc enai kai o zhtomenoc anagwgikc tpoc.

    q.e.d.

    6.8 Brete anagwgik tpo gia to oloklrwma x

    (x2+1)dx.

    Lsh: 'Estw J = x

    (x2+1)dx, tte J =

    x2x2+1

    x2dx all (x1) = ( 1)x2 x2 = (x

    1)

    1 antikajistntac, to J gnetai:

    J =1

    1

    x2x2 + 1

    (x1)dx

    diadoqik tra qoume

    J =1

    1

    [1 1

    x2 + 1

    ](x1)dx =

    =1

    1

    (x1)dx 1

    1

    (x1)x2 + 1

    dx =

    =x1

    1 1

    1

    ( 1) x

    2

    x2 + 1dx =

    =x1

    1 x2

    x2 + 1dx =

    x1

    1 J2

    o opooc enai kai o zhtomenoc anagwgikc tpoc.

    q.e.d.

    6.9 Brete anagwgik tpo gia to oloklrwma dx

    x

    x2+a, 6= 1, a 6=

    0.

  • 6.5. ASKHSEIS PROS EPILUSHN 41

    Lsh: 'Estw J = dx

    x

    x2+a, diadoqik qoume

    J2 = dx

    x2

    x2 + a= x

    x1

    x2 + adx =

    = (1

    x

    )1(

    x2 + a)dx =

    =

    x2 + a

    x1

    (

    x2 + a)(

    1

    x1

    )dx =

    =

    x2 + a

    x1+ ( 1)

    x2 + ax

    dx =

    =

    x2 + a

    x1+ ( 1)

    x2 + ax

    x2 + adx =

    =

    x2 + a

    x1+ ( 1)

    x2x

    x2 + adx + ( 1)

    adxx

    x2 + a

    kai telik qoume

    J2 =

    x2 + a

    x1+ ( 1)J2 + a( 1)J

    kai epilontac wc prc J , parnoume ton zhtomeno anagwgik tpo :

    J =

    x2 + a

    a( 1)x1

    a( 1)Jn2

    q.e.d.

    6.5 Askseic proc Eplushn

    6.10 Upologsate to oloklrwma

    7xdx

    Ap: 176xx 6

    354xx 24

    1052xx 48

    105+C

    6.11 Upologsate to oloklrwma 1

    (x2+7)2dx

  • 42 KEFALAIO 6. ANAGWGIKOI TUPOI

    Ap: x14(x2+7)

    + 114

    7o( x

    7) + C

    6.12 Upologsate to oloklrwma 1

    (3x2+5)2dx

    Ap: x10(3x2+5)

    + 110

    15o(

    35x) + C

    6.13 Upologsate to oloklrwma 1

    (x2x+7)3 dx

    Ap: 2x154(x2x+7)2 +

    2x1243(x2x+7) +

    4729

    3o

    (2x1

    3

    )6.14 Upologsate to oloklrwma

    1(7x2x+2)2 dx

    Ap: 14x155(7x2x+2) +

    2855

    55o

    (14x1

    55

    )

  • Keflaio 7

    Anlush se 'Ajroisma AplnKlasmtwn

    7.1 Genik

    To phlkon do akerawn poluwnmwn f(x) kai g(x) kaletai rht klsma rht sunrthsh wc proc x. Analutik grfoume:

    k(x) =f(x)

    g(x)=

    amxm + am1x

    m1 + + a1x + a0bnxn + bn1xn1 + + b1x + b0

    pou am, bn 6= 0 kai ai, bj ankoun sto R, i = 1, 2, . . . ,m, j = 1, 2, . . . , n.

    7.2 Anlush se Apl Klsmata

    Se pollc efarmogc qreizetai na analoume mia rht sunrthsh se jroi-sma aplosterwn klasmtwn. Akoloujome ton paraktw algrijmo.

    Algrijmoc anlushc se jroisma apln klasmtwn.

    Bma 1on. En o arijmhtc qei bajm megaltero tou paranomasto, k-noume thn diaresh f(x) : g(x). Qrhsimopointac thn tautthta thc

    diareshc qoume: f(x) = g(x)p(x) + v(x) kai ra f(x)g(x)

    = p(x) + v(x)g(x)

    ,

    pou to klsma v(x)g(x)

    qei plon arijmht bajmo mikrterou tou para-nomasto.

    Bma 2on. En o arijmhtc qei bajm mikrtero tou paranomasto, tteparagontopoiome plrwc ton paranomast.

    43

  • 44 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

    Bma 3on. Qrhsimopointac ton katllhlo tpo, exisnoume to f(x)g(x)

    mena jroisma apln klasmtwn pou periqoun stajerc proc prosdio-rismn.

    Bma 4on. Prosdiorzoume tic stajerc.

    7.3 Tpoi anlushc se apl klsmata

    H anlush sto 3on bma tou parapnw algorjmou, gnetai bsei twn tpwntwn ktwji periptsewn (Heavaside).

    Perptwsh I: En to g(x) qei mno aplc pragmatikc rzec p1, p2, . . . , pn,dhlad g(x) = (x p1)(x p2) (x pn) tte

    f(x)

    g(x)=

    f(x)

    (x p1)(x p2) (x pn)=

    A1(x p1)

    +A2

    (x p2)+ + An

    (x pn)pou A1, A2, . . . , An stajerc proc prosdiorismn.

    Perptwsh II: En to g(x) qei kai pollaplc rzec, dhlad g(x) = (x p1)(x p2)(x p3)k (x pm)l, tte

    f(x)

    g(x)=

    f(x)

    (x p1)(x p2)(x p3)k (x pm)l=

    =A1

    (x p1)+

    A2(x p2)

    +B1

    (x p3)+

    B2(x p3)2

    + + Bk(x p3)k

    +

    +M1

    (x pm)+

    M2(x pm)2

    + + Ml(x pm)l

    A1, A2, Bi, Mi stajerc proc prosdiorismn.

    Perptwsh III: En to g(x) qei thn morf g(x) = (1x2 + 1x + 1) (2x

    2 + 2x + 2) tte

    f(x)

    g(x)=

    f(x)

    (1x2 + 1x + 1) (2x2 + 2x + 2)=

    =Mx + N

    1x2 + 1x + 1+

    A1x + B12x2 + 2x + 2

    +

    +A2x + B2

    (2x2 + 2x + 2)2+ + Ax + B

    (2x2 + 2x + 2)

    Perptwsh IV : 'Otan isqoun sugqrnwc oi periptseic I, II, III, tteefarmzoume tautqrona touc antistoqouc tpouc.

  • 7.4. PROSDIORISMOS TWN STAJERWN 45

    7.4 Prosdiorismc twn stajern

    Gia na prosdiorsoume tic stajerc qrhsimopoiome do mejdouc: Ete k-noume ta klsmata omnuma kai exisnoume touc suntelestc twn omoiobaj-mwn rwn twn arijmhtn, ete knoume ta klsmata omnuma kai jtoumeaujaretec timc stic metablhtc x.

    7.5 Lumnec Askseic

    7.1 Na analuje to klsma

    x2 x 1(x 1)(x 2)(x + 3)

    se jroisma apln klasmtwn.

    Lsh: Akoloujntac ton algrijmo kai thn perptwsh I qoume:

    x2 x 1(x 1)(x 2)(x + 3)

    =A1

    x 1+

    A2x 2

    +A3

    x + 3=

    =A1(x 2)(x + 3) + A2(x 1)(x + 3) + A3(x 1)(x 2)

    (x 1)(x 2)(x + 3)exisnontac touc arijmhtc parnoume:

    x2 x 1 = A1(x 2)(x + 3) + A2(x 1)(x + 3) + A3(x 1)(x 2) (7.1)

    gia x = 1 h (7.1) gnetai: 1 = (4)A1 A1 = 1/4, gia x = 2 h (7.1) gnetai:1 = 5A2 A2 = 1/5, gia x = 3 h (7.1) gnetai: 11 = 20A3 A3 = 11/20kai h anlush telik enai:

    x2 x 1(x 1)(x 2)(x + 3)

    =1/4

    x 1+

    1/5

    x 2+

    11/20

    x + 3

    q.e.d.

    7.2 Na analuje to klsma

    x2 + x + 1

    (x + 2)(x + 3)2

    se jroisma apln klasmtwn.

  • 46 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

    Lsh: Apo thn perptwsh II qoume :

    x2 + x + 1

    (x + 2)(x + 3)2=

    A1x + 2

    +B1

    x + 3+

    B2(x + 3)2

    =

    =A1(x + 3)

    2 + B1(x + 2)(x + 3) + B2(x + 2)

    (x + 2)(x + 3)2

    exisnontac touc arijmhtc qoume:

    x2 + x + 1 = (A1 + B1)x2 + (6A1 + 5B1 + B2)x + (9A1 + 6B1 + 2B2)

    sugkrnontac touc suntelestc twn omoiobajmwn rwn, parnoume:

    A1 + B1 = 1

    6A1 + 5B1 + B2 = 1

    9A1 + 6B1 + 2B2 = 1

    kai raA1 = 3, B1 = 2, B2 = 7

    kai h anlush enai:

    x2 + x + 1

    (x + 2)(x + 3)2=

    3

    x + 2 2

    x + 3 7

    (x + 3)2

    q.e.d.

    7.3 Na analuje se jroisma apln klasmtwn to klsma:

    x4 + 1

    (x2 x + 1)3

    Lsh: Epeid qoume ston paranomast polunumo 2ou bajmo, ja qrhsi-mopoisoume ton tpo thc perptwshc III:

    x4 + 1

    (x2 x + 1)3=

    A1x + B1x2 x + 1

    +A2x + B2

    (x2 x + 1)2+

    +A3x + B3

    (x2 x + 1)3

    knontac ta klsmata omnuma qoume, exisnontac touc arijmhtc:

    x4 + 1 = (A1x + B1)(x2 x + 1)2 + (A2x + B2)(x2 x + 1) + (A3x + B3)

  • 7.5. LUMENES ASKHSEIS 47

    kai knontac prxeic:

    x4+1 = A1x5+(2A1+B1)x4+(3A1+A22B1)x3+(2A1A2+3B1+B2)x2+

    (A1 + A2 + A3 2B1 B2)x + (B1 + B2 + B3)

    Exisnontac touc suntelestc twn omoiobajmwn rwn parnoume to ssth-ma:

    A1 = 0

    2A1 + B1 = 1

    3A1 + A2 2B1 = 0

    2A1 A2 + 3B1 + B2 = 0

    A1 + A2 + A3 2B1 B2 = 0

    B1 + B2 + B3 = 1

    Ap pou brskoume ti:

    A1 = 0, A2 = 2, A3 = 1, B1 = 1, B2 = 1, B3 = 1

    Kai h anlush enai:

    x4 + 1

    (x2 x + 1)3=

    1

    x2 x + 1+

    2x 1(x2 x + 1)2

    +x + 1

    (x2 x + 1)3

    q.e.d.

    7.4 Na analuje se apl klsmata h rht sunrthsh:

    1

    (x2 + 1)(x2 + x)

    Lsh: Paragontopoiome plrwc ton paranomast kai qoume

    1

    (x2 + 1)(x2 + x)=

    1

    (x2 + 1)x(x + 1)

    Gia na analsoume se apl klsmata ja sundusoume lec tic periptseic:

    1

    (x2 + 1)(x2 + x)=

    A

    x+

    B

    (x + 1)+

    Gx + D

    x2 + 1

    Knontac ta klsmata omnuma kai exisnontac touc arijmhtc parnoume:

    1 = A(x + 1)(x2 + 1) + Bx(x2 + 1) + (Gx + D)x(x + 1) =

  • 48 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

    = (A + B + G)x3 + (A + G + D)x2 + (A + B + D)x + A

    ap pou brskoume ti:

    A = 1, B = 12, G = 1

    2, D = 1

    2

    kai h zhtoumnh anlush enai:

    1

    (x2 + 1)(x2 + x)=

    1

    (x2 + 1)(x2 + x)=

    1

    x 1

    2(x + 1) x + 1

    2(x2 + 1)

    q.e.d.

    7.5 Na analuje se apl klsmata to klsma

    x3

    x2 2x 3

    Lsh: Afo o arijmhtc qei bajm megaltero apo ton tou paranomasto,knoume diaresh kai qoume: x3 = (x+2)(x2 2x 3)+ (7x+6) to klsmaja gnei:

    x3

    x2 2x 3=

    (x + 2)(x2 2x 3) + 7x + 6x2 2x 3

    = (x + 2) +7x + 6

    x2 2x 3

    ja analsoume tra to klsma 7x+6x22x3 ,

    7x + 6

    x2 2x 3=

    7x + 6

    (x 3)(x + 1)=

    A

    x 3+

    B

    x + 1=

    =A(x + 1) + B(x 3)

    (x 3)(x + 1)exisnontac touc arijmhtc qoume : 7x+6 = A(x+1)+B(x 3), jtontacx = 1 kai x = 3, brskoume A = 27/4, B = 1/4, kai h anlush telikgnetai :

    x3

    x2 2x 3= (x + 2) +

    27/4

    x 3+

    1/4

    x + 1

    q.e.d.

    7.6 Upologsate to jroisma

    1

    1 3+

    1

    3 5+ + 1

    (2n 1)(2n + 1)

  • 7.5. LUMENES ASKHSEIS 49

    Lsh: Analoume to klsma 1(2n1)(2n+1) se jroisma apln klasmtwn:

    1

    (2n 1)(2n + 1)=

    A

    2n 1+

    B

    2n + 1=

    =A(2n + 1) + B(2n 1)

    (2n 1)(2n + 1) A = 1

    2, B = 1

    2

    kai ra1

    (2n 1)(2n + 1)=

    1/2

    2n 1+

    1/22n + 1

    Diadoqik tra qoume:

    n = 1 11 3

    =1

    2 1

    6

    n = 2 13 5

    =1

    6 1

    10

    n = 3 15 7

    =1

    10 1

    14

    n = n 1(2n 1)(2n + 1)

    =1/2

    2n 1+

    1/22n + 1

    Ajrozontac kat mlh, brskoume:

    1

    1 3+

    1

    3 5+ + 1

    (2n 1)(2n + 1)=

    1

    2 1

    2(2n + 1)

    q.e.d.

    7.7 Na analuje se jroisma apln klasmtwn to klsma

    2x2 + 3x 5(x 2)3

    Lsh: Ja lsoume thn skhsh aut me ma llh mjodo. Diairntac tonarijmht me to (x 2) qoume 2x2 + 3x 5 = (x 2)(2x + 7) + 9 all2x + 7 = 2(x 2) + 11 ra 2x2 + 3x 5 = 2(x 2)2 + 11(x 2) + 9 kaiepomnwc:

    2x2 + 3x 5(x 2)3

    =2(x 2)2 + 11(x 2) + 9

    (x 2)3=

    =2

    x 2+

    11

    (x 2)2+

    9

    (x 2)3

    q.e.d.

  • 50 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

    7.8 Na analuje se jroisma apln klasmtwn to klsma

    2x3 3x2 + 5x 3(x 1)5

    Lsh: Ja gryoume ton arijmht wc jroisma dunmewn tou (x 1), qrh-simopointac ton tpo tou Taylor. Gnwrzoume ti to anptugma Taylor mekntro to 1 enai:

    f(x) = f(1) + f (1)(x 1)

    1!+ f (1)

    (x 1)2

    2!+

    jtontac f(x) = 2x3 3x2 + 5x 3, parnoume:2x3 3x2 + 5x 3 = 1 + 5(x 1) + 3(x 1)2 + 2(x 1)3

    kai to klsma gnetai:

    2x3 3x2 + 5x 3(x 1)5

    =1 + 5(x 1) + 3(x 1)2 + 2(x 1)3

    (x 1)5=

    =1

    (x 1)5+

    5

    (x 1)4+

    3

    (x 1)3+

    2

    (x 1)2q.e.d.

    7.9 Na analuje to klsma 2x+1x4+1

    Lsh: Paragontopoiome ton paranomast:

    x4 + 1 = x4 + 2x2 2x2 + 1 = (x2 + 1)2 2x2 = (x2 + 1)2 (

    2x)2 =

    = (x2

    2x + 1)(x2 +

    2x + 1)

    kai to klsma analetai wc exc:

    2x + 1

    x4 + 1=

    Ax + B

    x2

    2x + 1+

    Gx + D

    x2 +

    2x + 1=

    =(Ax + B)(x2 +

    2x + 1) + (Gx + D)(x2

    2x + 1)

    x4 + 1Exisnontac touc suntelestc twn omoiobajmwn rwn twn arijmhtn, br-skoume ti :

    A = 12

    2, B =

    1

    2(1 +

    2), G =

    1

    2

    2, D =

    2 22

    2

    kai h anlush telik gnetai:

    12

    2x + 1

    2(1 +

    2)

    (x2

    2x + 1)+

    12

    2x +

    22

    2

    2

    (x2 +

    2x + 1)

    q.e.d.

  • 7.6. ASKHSEIS PROS EPILUSH 51

    7.6 Askseic proc Eplush

    Analsate se apl klsmata tic ktwji rhtc sunartseic

    7.10

    2x + 7

    (x2 4)(x + 1)

    Ap: 34(x+2)

    + 112(x2)

    53(x1)

    7.11

    3x 1x2 5x + 6

    Ap: 8x3

    5x2

    7.12

    x2 + 11x 2(x + 2)(x 1)(x 4)

    Ap: 19

    (29

    x4 10

    x1 10

    x+2

    )7.13

    10x2 + 32

    x3(x 4)2

    Ap: 1x

    + 1x2

    + 2x3 1

    x4 +3

    (x4)2

    7.14

    x5 + 2)

    (x2 + x + 1)3

    Ap: (x2)(x2+x+1)

    + x+3(x2+x+1)2

    + x1(x2+x+1)3

    7.15

    x2 x + 1(x2 + 1)(x 1)2

    Ap: 12(x2+1)

    + 12(x1)2

  • 52 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

    7.16

    x2

    (x2 2x + 5)2

    Ap: 1x22x+5 +

    2x5(x22x+5)2

    7.17

    5x2 4x4 5x2 + 4

    Ap: 13

    (4

    x2 1

    2(x1) +1

    2(x+1) 4

    x+2

    )7.18

    x3

    x3 3x + 2

    Ap: 1 + 13(x1)2 +

    89(x1)

    89(x+2)

    7.19

    2x3 2x 22x2 + x 6

    Ap: 12

    + x + 2x+2

    + 12(2x+3)

    7.20

    x + 1

    x4 5x3 + 9x2 7x + 2

    Ap: 3x2

    3x1

    3(x1)2

    2(x1)3

    7.21

    x2 x + 1(x 1)3

    Ap: 1x1 +

    1(x1)2 +

    1(x1)3

    7.22

    8

    x8 1

  • 7.6. ASKHSEIS PROS EPILUSH 53

    Ap: 1x1

    1x+1

    2x2+1

    4x4+1

    7.23

    x3

    (x + 1)5

    Ap: 1(x+2)2

    3(x+1)3

    + 3(x+1)4

    1(x+1)5

    7.24

    x4 1(3x + 1)2

    Ap: 127 2x

    27+ x

    2

    9 80

    81(3x+1)2 4

    81(3x+1)

    7.25

    2x2

    3x4 + 2x + 1

    Ap:

    7.26 Na breje to jroisma:

    1

    2 3+

    1

    3 4+ + 1

    (n + 1)(n + 2)

    Ap: 12 1

    n+2

    7.27 Na breje to jroisma:

    1

    1 3+

    1

    2 4+ + 1

    n(n + 2)

    Ap: 12[ 1n 1

    n+2

    7.28 Na breje to jroisma:

    1

    1 2 3+

    1

    2 3 4+ + 1

    n(n + 1)(n + 2)

    (Updeixh: Analsate to klsma 1n(n+1)(n+2)

    se jroisma klasmtwn me pa-

    ranomastc n(n + 1) kai (n + 1)(n + 2)).

    7.29 Dexate ti:

    1

    2 5+

    1

    5 8+ + 1

    (3n 1)(3n + 2)=

    1

    2

    (n

    3n + 2

    )

  • 54 KEFALAIO 7. ANALUSH SE AJROISMA APLWN KLASMATWN

  • Keflaio 8

    Basik Rht Oloklhrmata

    8.1 Genik

    Ta oloklhrmata twn ktwji rhtn sunartsewn qrhsimopoiontai gia tonupologism poluplokotrwn rhtn oloklhrwmtwn, all qoun kai autno-mh axa.

    O1 = dx

    (ax + b), O2 =

    dx(ax + b)n

    , O3 = dx

    ax2 + bx + c

    O4 = a1x + b1

    a2x2 + b2x + c2dx, O5 =

    dx(a1x2 + b1x + c1)n

    , O6 = a1x + b1

    (a1x2 + b1x + c1)ndx

    Sta tra teleutaa oloklhrmata h diakrnousa tou paranomasto jewretaiarnhtik. Ja antimetwpsoume to kje na xeqwrist.

    8.2 To oloklrwma O1

    To oloklrwma aut upologzetai bsei tou tpou:

    dxax + b

    =1

    aln |ax + b|+ C (8.1)

    Prosoq sthn perptwsh a < 0.

    8.3 To oloklrwma O2

    To oloklrwma aut upologzetai me antikatstash, jtontac w = ax + b.

    55

  • 56 KEFALAIO 8. BASIKA RHTA OLOKLHRWMATA

    8.4 To oloklrwma O3

    Upologzoume thn diakrnousa tou paranomasto kai meletme tic excperiptseic:

    Perptwsh 1h: > 0.

    Paragontopoiome ton paranomast ax2 + bx + c = a(x p1)(x p2) kaianaloume se jroisma apln klasmtwn: dx

    ax2 + bx + c= dx

    a(x p1)(x p2)=

    1

    a

    A(x p1)

    dx +1

    a

    B(x p2)

    dx

    Perptwsh 2h: = 0.

    Paragontopoiome ton paranomast ax2+bx+c = a(xp)2 kai met knoumeantikatstash, smfwna me thn perptwsh O2.

    Perptwsh 3h: < 0.Se autn thn perptwsh metatrpoume ton paranomast se jroisma tetra-gnwn: ax2 + bx + c = a[(x l)2 + d2] kai met qrhsimopoiome ton tpo touo, dhlad qoume: dx

    ax2 + bx + c= dx

    a[(x l)2 + d2]=

    =1

    a

    d(x l)(x l)2 + d2

    =1

    ado

    (x l

    d

    )+ C

    8.5 To oloklrwma O4

    Sqhmatzoume ston arijmht thn pargwgo tou paranomasto. Epeita dia-spme to klsma se duo klsmata kai epomnwc to arqik oloklrwma seduo oloklhrmata. To prto oloklrwma upologzetai me antikatstash,to detero ankei sthn kathgora O3.

    8.6 To oloklrwma O5

    Kat' arqc metatrpoume thn posthta a1x2 + b1x + c1 se jroisma tetra-gnwn: a1x2 + b1x + c1 = a1[(x s)2 + K2] kai met qrhsimopoiome tontpo:

    In =1

    K2

    (x s

    (2n 2)[(x s)2 + K2]n1+

    2n 32n 2

    In1

    ), n 6= 1

  • 8.7. TO OLOKLHRWMA O6 57

    pou In = 1

    [(xs)2+K2]n dx.

    PROSOQH: O arijmc a1 ja bge xw apo to oloklrwma.

    8.7 To oloklrwma O6

    Me katllhlec prxeic dhmiourgome ston arijmht thn pargwgo thc po-sthtac a1x2 + b1x + c1. Met diaspme to klsma se do klsmata kaiepomnwc to arqik oloklrwma se do oloklhrmata. To prto upolog-zetai me antikatstash, to detero ankei sthn kathgora O5.

    8.8 Lumnec Askseic

    8.1 Upologsate to dx

    2x+6.

    Lsh: Bsei tou tpou (8.1), qoume dx

    2x+6= 1

    2ln |2x + 6|+ C.

    q.e.d.

    8.2 Upologsate to dx

    5x .

    Lsh: Ap ton tpo (8.1) qoume, gia a = 1, dx

    5xdx = ln |5 x|+ C.

    q.e.d.

    8.3 Upologsate to dx

    (2x)100 .

    Lsh: Ja doulyoume me antikatstash. Jtoume w = 2 x kai radw = dx dx = dw. To oloklrwma tra gnetai: dx

    (2 x)100dx =

    dww100

    dw =

    w100dw =

    =w100+1

    (100 + 1)+ C = w

    99

    (99)+ C =

    1

    99 1(2 x)99

    + C

    q.e.d.

    8.4 Upologsate to oloklrwma dx(7 13x)13

  • 58 KEFALAIO 8. BASIKA RHTA OLOKLHRWMATA

    Lsh: Jtoume w = 7 13x, ra dw = 13dx dx = 113

    dw. Tooloklrwma gnetai: dx

    (7 13x)13=

    dw13w13

    dw = 113

    w13dw =

    = 113(

    w13+1

    13 + 1

    )+ C = 1

    13(

    w12

    12

    )+ C =

    =1

    156 1(7 13x)12

    + C

    q.e.d.

    8.5 Apodexate ton tpo: dx

    (ax+b)n= 1

    a(n1)(ax+b)n1 + C.

    Lsh: Me antikatstash qoume w = ax + b dw = adx dx = 1adw kai

    to oloklrwma gnetai. dx(ax + b)n

    = dw

    awn=

    1

    a

    wndw =

    =1

    a

    (wn+1

    n + 1

    )+ C = 1

    a(n 1)wn1+ C = 1

    a(n 1)(ax + b)n1)+ C

    q.e.d.

    8.6 Upologsate to dxx2 10x + 21

    Lsh: Blpoume ti = 16 > 0 kai epomnwc paragontopoiome ton pa-ranomast: x2 10x + 21 = (x 3)(x 7). Analoume tra to klsma

    1x210x+21 =

    1(x3)(x7) se jroisma apln klasmtwn. 'Htoi

    1

    x2 10x + 21=1/4x 3

    +1/4

    x 7

    kai to oloklrwma epomnwc gnetai: dxx2 10x + 21

    = dx

    (x 3)(x 7)= 1

    4

    dxx 3

    +1

    4

    dxx 7

    =

    = 14

    ln |x 3|+ 14

    ln |x 7|+ C = 14

    lnx 7x 3

    + Cq.e.d.

  • 8.8. LUMENES ASKHSEIS 59

    8.7 Upologsate to oloklrwma dx4x2 + 4x + 1

    Lsh: Parathrome ti = 0 kai o paranomastc grfetai 4x2 +4x+1 =(2x + 1)2, epomnwc gia to oloklrwma qoume: dx

    4x2 + 4x + 1= dx

    (2x + 1)2=

    12 dww2

    =1

    2

    (w2+1

    2 + 1

    )+ C = 1

    2w1 + C = 1

    2(2x + 1)+ C

    q.e.d.

    8.8 Upologsate to oloklrwma dx7x2 + 4x + 3

    Lsh: H diakrnousa tou paranomasto enai 68 < 0 kai epomnwc ja tonmetatryoume se jroisma tetragnwn. Diadoqik qoume:

    7x2 + 4x + 3 = 7(x2 + 4

    x

    7+

    3

    7

    )=

    = 7(x2 + 2 2

    7 x + 4

    49 4

    49+

    3

    7

    )= 7

    (x + 27

    )2+

    (17

    7

    )2To oloklrwma tra gnetai:

    dx7x2 + 4x + 3

    = 1/7[(

    x + 27

    )2+(

    177

    )2]dx =

    1

    7

    1177

    ox + 27

    177

    + C = 1717

    o

    (7x + 2

    17

    )+ C

    q.e.d.

    8.9 Upologsate to oloklrwma

    I = 7x + 9

    11x2 + x + 1dx

  • 60 KEFALAIO 8. BASIKA RHTA OLOKLHRWMATA

    Lsh: To oloklrwma ankei sthn kathgora O4. Sqhmatzoume ston a-rijmht thn pargwgo tou paranomasto (11x2 + x + 1) = 22x + 1 kai tooloklrwma mac gnetai:

    7x + 911x2 + x + 1

    dx =1227

    227 (7x + 9)

    11x2 + x + 1dx =

    =7

    22

    22x + 1987

    + 1 111x2 + x + 1

    dx =

    7

    22

    22x + 111x2 + x + 1

    dx +7

    22 191

    7

    111x2 + x + 1

    dx (8.2)

    To prto oloklrwma upologzetai me antikatstash, en to detero ankeisthn kathgora O3. Diadoqik qoume 22x + 1

    11x2 + x + 1dx =

    1w

    dw = ln |11x2 + x + 1|+ C (8.3)

    pou w = 11x2 + x + 1, dw = (22x + 1)dx.

    111x2 + x + 1

    dx =1

    11

    1x2 + 1

    11x + 1

    11

    dx =

    =1

    11

    1(x + 1

    22

    )2+(

    4322

    )2dx = 111 14322

    o

    x + 12243

    22

    + C =

    = 2

    43

    43o

    (22x + 1

    43

    )+ C (8.4)

    Antikajistntac ta (8.3) kai (8.4) sto (8.2) qoume telik:

    I =7

    22ln |11x2 + x + 1|+ 191

    473

    43o

    (22x + 1

    43

    )+ C

    q.e.d.

    8.10 Upologsate to oloklrwma

    I = dx

    (11x2 + x + 1)3

  • 8.8. LUMENES ASKHSEIS 61

    Lsh: Grfoume kat' arqc to oloklrwma sthn morf

    I = dx

    113(x2 + x

    11+ 1

    11

    )3 = 1113 dx(

    x2 + x11

    + 111

    )3Grfoume to x2 + x

    11+ 1

    11se jroisma tetragnwn x2 + x

    11+ 1

    11=(x + 1

    22

    )2+(

    4322

    )2kai to oloklrwma gnetai:

    dx(x2 + x

    11+ 1

    11

    )3 = dx((x + 1

    22

    )2+(

    4322

    )2)3 = dX

    (X2 + K2)3

    pou X = x + 1/22 kai K =

    13/22. Ja qrhsimopoisoume tra tonanadromik tpo thc perptwshc O5. To n = 3, ra

    dX(X2 + K2)3

    = I3 =1

    K2

    (X

    4(X2 + K2)2+

    3

    4I2

    )

    I2 =1

    K2

    (X

    2(X2 + K2)2+

    1

    2I1

    )

    I1 =1

    Ko

    X

    K

    Antikajistntac la ta prohgomena antstrofa kai lambnontac upyin tictimc twn X kai K brskoume telik to I.

    q.e.d.

    8.11 Upologsate to oloklrwma

    I = x 7

    (x2 x + 1)2dx

    Lsh: To oloklrwma aut ankei sthn kathgora O6. Ja sqhmatsoumeston arijmht thn pargwgo tou x2 x + 1, (x2 x + 1) = 2x 1. To Ignetai

    I =1

    2

    2(x 7)(x2 x + 1)2

    dx =1

    2

    2x 1 13(x2 x + 1)2

    dx =

    1

    2

    2x 1(x2 x + 1)2

    dx 132

    1(x2 x + 1)2

    dx (8.5)

  • 62 KEFALAIO 8. BASIKA RHTA OLOKLHRWMATA

    To prto oloklrwma upologzetai me antikatstash. Jtoume w = x2 x + 1 dw = (2x 1)dx kai ra 2x 1

    (x2 x + 1)2dx =

    1w2

    dw =

    w2dw =w1

    1+ C = 1

    x2 x + 1+ C

    (8.6)To detero oloklrwma ankei sthn kathgora O5. Grfoume to x2 x + 1se jroisma tetragnwn: x2 x + 1 =

    (x 1

    2

    )2+(

    32

    )2. To detero

    oloklrwma tra gnetai: 1(x2 x + 1)2

    dx = 1(

    x 12

    )2+(

    32

    )2dx = 1(X2 + K2)2dXpou X = x 1

    2, K =

    3

    2. Qrhsimopointac tra ton tpo thc O5 qoume: 1

    (X2 + K2)2dX = I2 =

    1

    K2X

    2(X2 + K2)+

    1

    2I1

    I1 =1

    Ko

    (X

    K

    )+ C =

    23o

    (2x 1

    3

    )+ C

    1(x2 x + 1)2

    dx =4

    3

    [x 1/2

    2[(x 1/2)2 + 3/4]

    ]

    +1

    2

    23o

    (2x 1

    3

    )+ C (8.7)

    Antikajistntac ta apotelsmata (8.6),(8.7), sto (8.5) kai knontac tic pr-xeic brskoume telik :

    I =1

    2(x2 x + 1) 13 (x 1/2)

    [(x 1/2)2 + (3/4)] 26

    3

    3o

    (2x 1)3

    + C

    q.e.d.

    8.9 Askseic Proc Eplushn.

    8.12 Upologsate to oloklrwma dx

    3x+7.

    Ap: 13ln |3x + 7|+ C.

    8.13 Upologsate to dx

    128x .

  • 8.9. ASKHSEIS PROS EPILUSHN. 63

    Ap: 18ln |12 8x|+ C.

    8.14 Apodexate ton tpo thc perptwshc O5.

    8.15 Upologsate to dx

    (2x9)20 .

    Ap: 138 1

    (2x9)19 + C.

    8.16 Upologsate to dx

    (105x)123 .

    Ap: 15 1

    122 1

    (105x)122 + C.

    8.17 Upologsate to dx

    4x28x .

    Ap: 18lnx2

    x

    + C.8.18 Upologsate to

    dxx29 .

    Ap: 16lnx3x+3

    + C.8.19 Upologsate to

    dxa(x2A) .

    Ap: 1a

    AlnxAx+

    A

    + C.8.20 Upologsate to

    dx2x26x7 .

    Ap: 12

    23ln2x3232x3+

    23

    + C.8.21 Upologsate to

    dxx26x+9 .

    Ap: 1x3 + C.

    8.22 Upologsate to dx

    x24x+4 .

    Ap: 1x2 + C.

    8.23 Upologsate to dx

    x2+2

    2x+2.

    Ap: 1x+

    2

    + C.

    8.24 Upologsate to dx

    x26x+13 .

  • 64 KEFALAIO 8. BASIKA RHTA OLOKLHRWMATA

    Ap: 12o

    (x3

    2

    )+ C.

    8.25 Upologsate to dx

    2x2+3x+4.

    Ap: 2

    2323

    o(

    4x+323

    )+ C.

    8.26 Upologsate to dx

    x2x+1 .

    Ap: 2

    33

    o(

    2x13

    )+ C.

    8.27 Upologsate to dx

    x28x+17 .

    Ap: o(x 4) + C.

    8.28 Upologsate to 7x+8

    4x216x+52dx.

    Ap: 78ln |4x216x+52|+11

    6o

    (x2

    3

    )+

    C.

    8.29 Upologsate to 2x3

    4x2+11dx.

    Ap: 14ln |4x2 + 11| 3

    2

    11o

    (2x11

    )+ C.

    8.30 Upologsate to dx

    (x2+2)3.

    Ap: x8(x2+2)2

    + 3x32(x2+2)

    + 332

    2o

    (x2

    )+ C.

    8.31 Upologsate to 3x

    (x2+7)2dx.

    Ap: 32(x2+7)

  • Keflaio 9

    Oloklrwsh Rhtn Sunartsewn

    9.1 Genik

    Jloume na upologsoume oloklhrmata thc morfc:

    P (x)Q(x)

    dx (9.1)

    pou P (x) kai Q(x) polunumo tou x. Autc oi sunartseic lgontai rhtckai ta antstoiqa oloklhrmata rht.

    9.2 H Mjodoc

    Gia ton upologism tou (9.1) akoloujome thn exc mjodo:

    BHMA 1on: En o bajmc tou P (x) enai megalteroc tou bajmo touQ(x), knoume thn diaresh kai diaspme to arqik oloklrwma (9.1),se na poluwnumik oloklrwma kai se na oloklrwma rhtc sunr-thshc me bajm arijmhto, mikrtero tou bajmo tou paranomasto.

    BHMA 2on: En o bajmc tou arijmhto enai mikrteroc tou bajmo touparanomasto, paragontopoiome plrwc ton paranomast.

    BHMA 3on: Analoume to klsma P (x)Q(x)

    se jroisma apln klasmtwn.

    BHMA 4on: Diaspme to arqik oloklrwma (9.1) se epimrouc basikrht oloklhrmata, ta opoa upologzontai kata ta gnwst.

    65

  • 66 KEFALAIO 9. OLOKLHRWSH RHTWN SUNARTHSEWN

    9.3 Lumnec Askseic

    9.1 Upologsate to dxx3 1

    Lsh: Paragontopoiome kat' arqc ton paranomast: x3 1 = (x 1)(x2 +x+1) kai analoume to klsma 1

    x31 se jroisma apln klasmtwn:

    1

    x3 1=

    1

    (x 1)(x2 + x + 1)=

    A

    x 1+

    Bx + G

    x2 + x + 1

    Me prxeic brskoume oti

    A =1

    3, B = 1

    3, G = 2

    3

    kai to oloklrwma gnetai: dxx3 1

    =1

    3 dx

    x 1 1

    3 x + 2

    x2 + x + 1dx

    Upologzoume ta epimrouc oloklhrmata. To prto enai: dxx 1

    = ln |x 1|+ C

    To detero ankei sthn kathgora O4 kai sqhmatzoume ston arijmht thnpargwgo tou paranomasto: x + 2

    x2 + x + 1dx =

    1

    2

    2x + 4x2 + x + 1

    dx =1

    2

    2x + 1 + 3x2 + x + 1

    dx =

    =1

    2

    2x + 1x2 + x + 1

    dx +3

    2

    dxx2 + x + 1

    =

    =1

    2

    2x + 1x2 + x + 1

    dx +3

    2

    dx[(x + 1

    2)2 + (

    3

    2)]2

    =

    =1

    2ln |x2 + x + 1|+ 3

    2

    (23

    )o

    x + 123

    2

    + CAntikajistntac qoume telik:

    dxx3 1

    =1

    3ln |x 1| 1

    6ln |x2 + x + 1|

    3

    3o

    (2x + 1

    3

    )+ C

    q.e.d.

  • 9.3. LUMENES ASKHSEIS 67

    9.2 Upologsate to dxx3 19x + 30

    Lsh: Paragontopoiome kat' arqc ton paranomast: x3 19x + 30 =(x 2)(x 3)(x + 5) kai to oloklrwma gnetai: dx

    x3 19x + 30= dx

    (x 2)(x 3)(x + 5)

    Analoume se jroisma apln klasmtwn:

    1

    (x 2)(x 3)(x + 5)=

    A

    x 2+

    B

    x 3+

    G

    x + 5

    kai upologzoume oti:

    A = 17, B =

    1

    8, G =

    1

    56

    kai to oloklrwma metatrpetai se dxx3 19x + 30

    = 17

    dxx 2

    +1

    8

    dxx 3

    +1

    56

    dxx + 5

    =

    = 17

    ln |x 2|+ 18

    ln |x 3|+ 156

    ln |x + 5|+ C =

    =1

    56ln

    (x 3)7(x + 5)(x 2)8+ C

    q.e.d.

    9.3 Upologsate to oloklrwma: x2 + 6x 1x4 + x

    dx

    Lsh: Paragontopoiome plrwc ton paranomast: x4 + x = x(x3 + 1) =x(x+1)(x2x+1). Analoume tra to klsma x2+6x1

    x4+xse jroisma apln

    klasmtwn:

    x2 + 6x 1x4 + x

    =x2 + 6x 1

    x(x + 1)(x2 x + 1)=

    A

    x+

    B

    x + 1+

    Gx + D

    x2 x + 1

    Me exswsh twn suntelestn twn omoiobajmwn rwn brskoume

    A = 1, B = 2, G = 1, D = 4

  • 68 KEFALAIO 9. OLOKLHRWSH RHTWN SUNARTHSEWN

    kai to arqik klsma gnetai:

    x2 + 6x 1x4 + x

    = 1x

    +2

    x + 1+

    x + 4x2 x + 1

    To arqik oloklrwma tra gnetai:

    x2 + 6x 1x4 + x

    dx = dx

    x+ 2

    dxx + 1

    + x + 4

    x2 x + 1dx

    Ta epimrouc oloklhrmata enai:

    dxx

    = ln |x|+ C1

    dxx + 1

    = ln |x + 1|+ C2 x + 4

    x2 x + 1dx =

    x 4x2 x + 1

    dx =

    = (

    1

    2

    ) 2x 8x2 x + 1

    dx = 12

    2x 1 7x2 x + 1

    dx =

    = 12

    2x 1x2 x + 1

    dx +7

    2

    dx(x 1/2)2 + (

    3/2)

    =

    = 12

    ln |x2 x + 1|+ 72

    1

    32

    o

    x 1/23

    2

    + C3Me antikatstash kai ektlesh prxewn brskoume to telik apotlesma:

    x2 + 6x 1x4 + x

    dx = ln |x|+ 2 ln |x + 1| 12

    ln |x2 x + 1|+

    +7

    2

    1

    32

    o

    x 1/23

    2

    + Cq.e.d.

    9.4 Upologsate to oloklrwma:

    x3x2 + 1

    dx

  • 9.4. ASKHSEIS PROS EPILUSHN. 69

    Lsh: Epeid o arijmhtc qei bajm megaltero tou paranomasto, jaknoume thn diaresh: x3 : x2 + 1 kai brskoume phlko x kai uploipo x,epomnwc x3 = (x2 + 1)x x. Antikajistntac aut thn kfrash stonarijmht, to oloklrwma gnetai:

    x3x2 + 1

    dx = (x2 + 1)x x

    x2 + 1dx =

    xdx

    xx2 + 1

    dx =

    =x2

    2 1

    2

    2xx2 + 1

    dx =x2

    2 1

    2ln |x2 + 1|+ C

    q.e.d.

    9.5 Upologsate to oloklrwma:

    x2 + 2x 1x2 + 1

    dx

    Lsh: Epeid o arijmhtc qei ton dio bajm me ton bajm tou paranoma-sto, ja knoume thn diaresh. Brskoume phlko son me 1 kai uploipo some 2x 2, epomnwc x2 + 2x 1 = (x2 + 1)1 + (2x 2) kai to oloklrwmagnetai:

    x2 + 2x 1x2 + 1

    dx = (x2 + 1) 1 + (2x 2)

    x2 + 1dx =

    =

    1dx + 2x 1

    x2 + 1dx = x +

    2xx2 + 1

    dx 2 1

    x2 + 1dx =

    = x + ln |x2 + 1| 2ox + C

    q.e.d.

    9.4 Askseic proc Eplushn.

    Upologsate ta ktwji oloklhrmata:

    9.6 2x

    5x2+4x7dx

    Ap: 1770

    ln |10x 3| 3170

    ln |10x + 11|+ C

    9.7 x27

    x+2dx.

    Ap: 3 ln |x + 2|+ x22 2x + C

  • 70 KEFALAIO 9. OLOKLHRWSH RHTWN SUNARTHSEWN

    9.8 2x4

    (x2+1)2dx.

    Ap: 2x + xx2+1

    3ox + C

    9.9 dx

    x3+1.

    Ap: 13ln |x+1| 1

    6ln |x2x+1|+ 1

    3o

    (2x1

    3

    )+C

    9.10 dx

    x31 .

    Ap: 13ln |x1| 1

    6ln |x2 +x+1| 1

    3o

    (2x+1

    3

    )+C

    9.11 x4x3x1

    x2x dx.

    Ap: x3

    3+ ln |x| 2 ln |x 1|+ C

    9.12 dx

    x4+q4.

    Ap: 12

    2q3

    [o

    (2x

    2q

    2q

    )+ o

    (2x+

    2q

    2q

    )]+ 1

    4

    2q3ln q2+2qx+x2q2

    2qx+x2

    +C

    9.13 dx

    x4+x2+1.

    Ap: 12

    3

    [o

    (2x1

    3

    )+ o

    (2x+1

    3

    )]+ 1

    4lnx2+x+1x2x+1

    + C9.14

    x5dxx3+x

    Ap: x3

    x xox + C

    9.15 x2+5x6

    x27x+9dx.

    Ap:

    9.16 5x6

    x+7dx

    Ap: 5x 41 ln |x + 7|+ C

    9.17 x2+2

    (x+1)3(x2)dx.

    Ap: 12x6(1+x)2

    + 29lnx2x+7

    + C9.18

    x37x2+11(x2)2(x+1)2 dx

    Ap: 1x2

    13(x+1)

    109

    ln |x 2|+ 199

    ln |x + 1|+ C

    9.19 2x

    (x2+1)(x2+3)dx

    Ap: 12lnx2+1x2+3

    + C

  • Keflaio 10

    Oloklrwsh TrigwnometriknSunartsewn

    10.1 Genik

    Parti uprqei genikc trpoc gia thn oloklrwsh twn trigwnometriknsunartsewn, ton opoo kai ja exetsoume sto tloc tou kefalaou, merikctrigwnometrikc sunartseic oloklhrnontai eukoltera. Ja xekinsoumeto keflaio me thn parousish autn twn eidikn morfn.

    10.2 Trigwnometrik Ginmena

    Metatrpoume ta ginmena se ajrosmata, bsei twn tpwn:

    (Ax)(Bx)dx =

    1

    2

    (Ax + Bx)dx +

    1

    2

    (AxBx)dx

    (Ax)(Bx)dx =

    1

    2

    (Ax + Bx)dx 1

    2

    (AxBx)dx

    (Ax)(Bx)dx =

    1

    2

    (Ax + Bx)dx +

    1

    2

    (AxBx)dx

    (Ax)(Bx)dx =

    1

    2

    (AxBx)dx 1

    2

    (Ax + Bx)dx

    71

  • 72KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

    10.3 Oloklrwsh perittn dunmewn, trigwno-metrikn sunartsewn

    Isqoun oi ktwji empeiriko kannec:

    To oloklrwma

    2n+1xdx upologzetai me thn antikatstash w =

    x.

    To oloklrwma

    2n+1xdx upologzetai me thn antikatstash

    w = x.

    To oloklrwma

    2n+1xdx upologzetai me thn metatrop x =

    xx kai thn antikatstash w = x.

    To oloklrwma

    2n+1xdx upologzetai me thn metatrop x =

    xx kai thn antikatstash w = x.

    10.4 Oloklrwsh artwn dunmewn, trigwno-metrikn sunartsewn

    Isqoun oi ktwji empeiriko kannec :

    Gia na upologsoume to oloklrwma

    2xdx qrhsimopoiome to-

    n tpo: 2x = 12x2

    , gia na proume oloklhrmata mikrotroubajmo.

    Gia na upologsoume to oloklrwma

    2xdx qrhsimopoiome ton

    tpo: 2x = 1+2x2

    gia na proume oloklhrmata mikrotroubajmo.

    Gia na proume to oloklrwma

    2xdx metatrpoume to 2x se

    ginmeno 22x2x kai qrhsimopoiome ton tpo 2x = 12x1.Prokptoun do oloklhrmata, to prto upologzetai me thn antikat-stash w = x kai to detero enai moio me to arqik all mikrotroubajmo.

  • 10.5. OLOKLHRWMATA GINOMENWNDUNAMEWN TRIGWNOMETRIKWN SUNARTHSEWN73

    Gia na proume to oloklrwma

    2xdx paragontopoiome 2x

    smfwna me thn sqsh 22x 2x kai qrhsimopoiome ton tpo2x = 12x 1. Prokptoun do oloklhrmata, to prto upolog-zetai me thn antikatstash w = x kai to detero enai moio me toarqik all mikrotrou bajmo.

    10.5 Oloklhrmata ginomnwn dunmewn tri-gwnometrikn sunartsewn

    Isqoun oi ktwji empeiriko kannec:

    Gia na upologsoume oloklhrmata thc morfc

    x2k+1xdx ,

    , k akraioi, ekfrzoume to oloklrwma sunartsei tou x kai qrh-simopoiome thn antikatstash w = x

    Gia na upologsoume oloklhrmata thc morfc

    x2kxdx , , k

    akraioi, ekfrzoume to oloklrwma sunartsei tou x kai qrhsi-mopoiome thn antikatstash w = x.

    10.6 Oloklhrmata phlkwn dunmewn trigw-nometrikn sunartsewn

    Gia na upologsoume oloklhrmata thc morfc x

    xdx , -

    jetiko akraioi, ta metatrpoume, qrhsimopointac ton tpo 2x +2x = 1, se oloklrwma enc mno trigwnometriko arijmo kaidiaspme to klsma.

    Gia na upologsoume oloklhrmata thc morfc 1

    xxdx ,

    - jetiko akraioi, antikajistome thn monda tou arijmhto me thnposthta 2x + 2x = 1 kai diaspme to klsma.

  • 74KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

    10.7 Genikc trpoc oloklrwshc trigwnome-trikn sunartsewn

    'Ena opoiodpote trigwnometrik oloklrwma thc morfc

    f(x, x, x)dx

    metatrpetai se rht, me thn bojeia twn metasqhmatismn:

    dx =2dt

    t2 + 1, x =

    2t

    1 + t2, x =

    1 t2

    1 + t2, x =

    2t

    1 t2

    Oi metasqhmatismo auto prokptoun apo thn sqsh t = x2. H mjodoc

    aut, parti antimetwpzei opoiodpote trigwnometrik oloklrwma, odhgesunjwc se perploka rht oloklhrmata kai gia aut kal enai na apo-fegetai.

    10.8 Lumnec Askseic

    10.1 Upologsate to oloklrwma

    (5x)(2x)dx.

    Lsh: Ap touc gnwstoc trigwnometrikoc tpouc qoume:

    (5x)(2x) =1

    2(5x + 2x) +

    1

    2(5x 2x) =

    =1

    2(7x) +

    1

    2(3x)

    kai ra (5x)(2x)dx =

    1

    2

    [(7x)dx +

    (3x)dx

    ]=

    =1

    14(7x) +

    1

    6(3x) + C

    q.e.d.

    10.2 Upologsate to oloklrwma

    3xdx.

    Lsh: Jtoume w = x kai ra dwdx

    = x dx = dwx . Diadoqiktra qoume:

    3xdx =

    3x1

    xdw =

    =

    2xdw =

    (1 2x)dw =

    (1 w2)dw =

  • 10.8. LUMENES ASKHSEIS 75

    (1)dw

    w2dw = w w

    3

    3+ C = x

    3x

    3+ C

    q.e.d.

    10.3 Upologsate to oloklrwma

    5xdx.

    Lsh: Ja qrhsimopoisoume thn antikatstash w = x. Diadoqikqoume:

    5xdx = 5x

    5xdx =

    5xw5

    (1x

    )dw =

    = 4x

    w5dw =

    (1 2x)2w5

    dw = (1 w2)2

    w5dw =

    = dw

    w5 2

    dww3

    + dw

    wdw =

    1

    4w4 1

    w2+ ln w + C =

    =1

    44x 1

    2x+ ln |x|+ C

    q.e.d.

    10.4 Upologsate to oloklrwma

    4xdx.

    Lsh: Diadoqik qoume:

    4xdx =

    (2x)2dx =

    (1 + (2x)2

    )2dx =

    =1

    4

    [1 + 2(2x) + 2(2x)]dx =

    1

    4

    1dx +

    2

    4

    (2x)dx+

    +1

    4

    2(2x)dx =

    x

    4+

    1

    4 12 (2x) + 1

    4

    1 + (4x)2

    dx =

    =x

    4+

    1

    4(4x) +

    1

    8

    1dx +

    1

    8

    (4x)dx =

    =x

    4+

    1

    4(4x) +

    x

    8+

    1

    32(4x) + C

    q.e.d.

    10.5 Upologsate to oloklrwma

    4xdx.

  • 76KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

    Lsh: Diadoqik qoume:

    4xdx =

    2x 2xdx =

    2x

    (1

    2x 1

    )dx =

    = 2x

    2xdx

    2xdx

    To prto oloklrwma upologzetai me antikatstash w = x dw = dx2x kai ra: 2x

    2xdx =

    (w2)dw = w

    3

    3+ C1 =

    3x

    3+ C1

    Gia to detero oloklrwma qoume:

    2xdx =

    ( 12x

    1)

    dx = dx

    2x

    1dx = x x + C2

    kai telik 4xdx =

    3x

    3 x x + C

    q.e.d.

    10.6 Upologsate to oloklrwma

    3x2xdx.

    Lsh: Qrhsimopoiome thn antikatstash w = x kai qoume dx = dwxkai ra

    3x 2xdx =

    2x x 2xdx =

    =

    (1 2x) x 2x ( 1

    x

    )dw =

    (1 w2)w2dw =

    (w4 w2)dw = w5

    5 w

    3

    3+ C =

    =1

    55x 1

    33x + C

    q.e.d.

    10.7 Upologsate to oloklrwma

    5x2x

    dx

  • 10.8. LUMENES ASKHSEIS 77

    Lsh: Qrhsimopoiome thn antikatstash w = x dx = dwx kaira 5x

    2xdx =

    5xw2

    ( 1

    x

    )dw =

    = 4x

    w2dw =

    (1 2x)2w2

    dw =

    = (1 w2)2

    w2dw =

    (1 2w2 + w4w2

    )dw =

    = 1

    w2dw + 2

    (1)dw

    w2dw =

    =1

    w+ 2w w

    3

    3+ C =

    1

    x+ 2x

    3x

    3+ C

    q.e.d.

    10.8 Upologsate to oloklrwma dx2x2x

    Lsh: Sqhmatzoume ston arijmht thn posthta 2x + 2x = 1 kaiqoume: 1

    2x2xdx =

    2x + 2x2x2x

    dx =

    = 2x

    2x2xdx +

    2x2x2x

    dx =

    = dx

    2x+ dx

    2x= x x + C

    q.e.d.

    10.9 Upologsate to oloklrwma dx1 + x

    Lsh: Ja qrhsimopoisoume thn mjodo thc genikc trigwnometrikc anti-katstashc. Jtoume x = 1t

    2

    1+t2, dx = 2dt

    1+t2, pou t = (x

    2) kai qoume:

    dx2 + x

    = 2

    1+t2

    5 + 1t2

    1+t2

    dt =

  • 78KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

    = 2

    6 + 4t2dt =

    dt2t2 + 3

    to teleutao oloklrwma ja gnei: dt2t2 + 3

    dt =1

    2

    dtt2 + 3

    2

    dt =

    =1

    2

    dtt2 + (

    32)2

    =1

    2 1

    32

    o

    t32

    + C ==

    2

    2

    3o

    (t

    23

    )+ C

    kai me antstrofh antikatstash qoume telik:

    dx1 + x

    =

    2

    2

    3o

    (

    x2

    )2

    3

    + Cq.e.d.

    10.10 Upologsate to oloklrwma dx2 + 3x + x

    Lsh: Me antikatstash qoume:

    21+t2

    2 + 3 2tt2+1

    + 1t2

    1+t2

    dt =

    = 2 dt

    2t2 + 2 + 6t + 1 t2= dt

    t2 + 6t + 3

    Analoume to klsma 1t2+6t+3

    se jroisma apln klasmtwn:

    1

    t2 + 6t + 3= 1

    2

    6(3 +

    6 t) 1

    2

    6(t + 3 +

    6)

    kai to oloklrwma gnetai:

    2 dt

    t2 + 6t + 3= 1

    6

    dt3 +

    6 t

    16

    dtt + 3 +

    6

    =

    16

    ln | 3 +

    6 t| 16

    ln |t + 3 +

    6|+ C =

  • 10.9. ASKHSEIS PROS EPILUSHN 79

    =16

    ln

    3 +

    6 tt + 3 +

    6

    + Ckai me antstrofh antikatstash qoume telik:

    dx2 + 3x + x

    =16

    ln

    3 +

    6 (x2)

    (x2) + 3 +

    6

    + Cq.e.d.

    10.9 Askseic proc Eplushn

    Upologsate ta ktwji oloklhrmata:

    10.11

    (x2)(x

    2)dx

    Ap: 35(5

    6x) 3(x

    6) + C

    10.12

    x(nx)dx

    Ap: 12(n+1)

    ((n + 1)x) + 12(n+1)

    ((1 n)x) + C

    10.13

    (10x)(6x)dx

    Ap: 18(4x) + 1

    32(16x) + C

    10.14

    x(nx)dx

    Ap: 12(n+1)

    ((n + 1)x) + 12(1n)((1 n)x) + C

    10.15

    3xdx

    Ap: ln |x|+ 122x + C

    10.16

    4xdx

    Ap: 133x x + x + C

    10.17

    6xdx

    Ap: 155x + 1

    33x x x + C

    10.18 dx

    23x

  • 80KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

    Ap: x2x +

    12ln 1x + x+ dx3x

    10.19 dx

    54x

    Ap: 23o(3(x

    2)) + C

    10.20 dx

    x

    Ap: ln 1x x+ C

    10.21 dx

    x

    Ap: ln 1x + x+ C

    10.22 dx

    3x

    Ap: x2x +

    12ln 1x x+ C

    10.23 dx

    3x

    Ap: x2x +

    12ln 1x + x+ C

    10.24 3x

    4(3x)dx

    Ap: 162(3x) 1

    124(3x) + C

    10.25 dx

    1(x2)

    Ap: 2(

    x2

    + 1(x2)

    )+ C

    10.26 dx

    1+xx

    Ap: ln (x2 )1+(x

    2)

    + C10.27

    dx1+(3x)

    Ap: 1(3x)3(3x) + C

    10.28 dx

    12x

  • 10.9. ASKHSEIS PROS EPILUSHN 81

    Ap:

    33

    ln(x2 )23(x

    2)2+

    3

    + C10.29

    x1+2xdx

    Ap:

    24

    ln2(x2 )+322)2(x

    2)+3+2

    2

    + C10.30

    dx2+xdx

    Ap: 23o

    (2(x

    2+1)

    3

    )+ C

  • 82KEFALAIO 10. OLOKLHRWSH TRIGWNOMETRIKWN SUNARTHSEWN

  • Keflaio 11

    Oloklrwsh ArrtwnSunartsewn I

    Sto keflaio aut ja asqolhjome me thn oloklrwsh arrtwn sunartse-wn me prwtobjmio uprrizo.

    11.1 Oloklhrmata thc morfc:

    f

    (x, n

    x +

    x +

    )dx ,

    6=

    Ta oloklhrmata aut upologzontaai me thn antikatstash = n

    x+x+

    ,metatrepmena se rht.

    11.2 Oloklhrmata thc morfc:

    f

    (x, n1

    x +

    x + , n2

    x +

    x +

    )dx ,

    6=

    Gia na upologsoume aut ta oloklhrmata, metatrpoume tic eterobjmiecrzec omoiobjmiec, me thn bojeia tou elaqstou koino pollaplasou n twn

    n1, n2 kai met qrhsimopoiome thn antikatstash = n

    x+x+

    . H antikat-stash aut metasqhmatzei to arqik oloklrwma se rht.

    83

  • 84 KEFALAIO 11. OLOKLHRWSH ARRHTWN SUNARTHSEWN I

    11.3 Oloklhrmata thc morfc:

    f(x, n1

    xm1 , n2

    xm2 , . . . , nk

    xmk

    )dx

    Gia na upologsoume aut ta oloklhrmata metatrpoume tic eternumecrzec se omnumec, me thn bojeia tou elaqstou koino pollaplasou n twntxewn twn rizn n1, n2, . . . , nk kai met qrhsimopoiome ton metasqhmatismn

    x = , gia na metatryoume to oloklrwma se rht.

    11.4 Lumnec Askseic

    11.1 Upologsate to oloklrwma: dx(x + 2)

    x 7

    Lsh: Qrhsimopoiome thn antikatstash:

    x 7 = kai qoume diado-qik: x = 2 + 7 dx = 2d, kai to oloklrwma gnetai: dx

    (x + 2)

    x 7= 2d

    (2 + 7 + 2)=

    = 2 d

    2 + 9= 2

    d2 + 32

    =

    = 2 13o

    (

    3

    )+ C =

    2

    3o

    (x 73

    )+ C

    q.e.d.

    11.2 Upologsate to oloklrwma: 2x 33

    x 2dx

    Lsh: Jtoume 3

    x 2 = kai qoume: x = 3 + 2 kai dx = 32d. Tooloklrwma tra diadoqik gnetai: 2x 3

    3

    x 2dx =

    2(3 + 2) 3

    32d =

    3(23 + 1)d =

    =

    (64 + 3)d =6

    55 +

    3

    22 + C =

    6

    53

    (x 2)5 + 3

    23

    (x 2)2 + C

    q.e.d.

  • 11.4. LUMENES ASKHSEIS 85

    11.3 Upologsate to oloklrwma:

    1x(x 1)

    3

    x

    x 1dx

    Lsh: Ja qrhsimopoisoume thn antikatstash: 3

    xx1 = ap' pou qoume

    x = 3

    31 kai me paraggish parnoumedxd

    = 32

    (31)2 dx =32

    (31)2 d. Tooloklrwma tra diadoqik gnetai:

    1x(x 1)

    3

    x

    x 1dx =

    13

    31

    (3

    31 1) 32

    (3 1)2d =

    = 33

    3d = 3

    d = 3 + C = 3 3

    x

    x 1+ C

    q.e.d.

    11.4 Upologsate to oloklrwma:

    I = 1 3x + 1

    x + 1 + 3

    x + 1dx

    Lsh: Metatrpoume ta eterobjmia rizik se omoiobjmia kai to olokl-rwma gnetai:

    I = 1 6(x + 1)2

    6

    (x + 1)6 + 6

    (x + 1)2

    dx

    Qrhsimopoiome tra thn antikatstash = 6

    x + 1, ap pou x = 6 1kai dx = 65d. 'Eqoume tra diadoqik:

    I = 1 2

    3 + 2 65d =

    = ( 1)( + 1)

    2( + 1) 65d = 6

    ( 1)2d =

    = 6

    (4 3)d = 655 +

    6

    44 + C =

    = 65( 6

    x + 1)5 +3

    2( 6

    x + 1)4 + C

    q.e.d.

  • 86 KEFALAIO 11. OLOKLHRWSH ARRHTWN SUNARTHSEWN I

    11.5 Upologsate to oloklrwma:

    I = x

    4

    x3 + 1dx

    Lsh: Metatrpoume ta eterobjmia rizik se omoiobjmia kai qoume:

    I = 4x2

    4

    x3 + 1dx

    Jtoume tra = 4

    x, dx = 43d kai to oloklrwma diadoqik gnetai:

    I = 2

    3 + 1 43d = 4

    53 + 1

    d =

    = 4 (

    frac23 + 1)d = 4

    d 4

    frac23 + 1d =

    =4

    34

    x3 43

    ln | 4

    x3 + 1|+ C

    q.e.d.

    11.5 Askseic proc Eplush

    Upologsate ta ktwji oloklhrmata:

    11.6 dx

    3x5

    Ap: 23

    3x 5 + C.

    11.7 13x+2

    1+

    3x+2dx

    Ap: 13(3x + 2) + 4

    3

    3x + 2 4

    3ln |

    3x + 2 +1|+ C.

    11.8 1

    x

    1x1+x

    dx

    Ap: 2o(

    1x1+x

    )+ ln

    1x1+x1x+

    1+x

    + C.11.9

    x2 3x24x dx

    Ap: 45x15/12 24

    17x17/12 + C.

  • 11.5. ASKHSEIS PROS EPILUSH 87

    11.10 dx

    x

    1x

    Ap: ln11x1+

    1x

    + C.11.11

    dx3+

    x+2

    Ap: 2

    x + 2 6 ln(3 +

    x + 2) + C.

    11.12 3x+2

    x3

    Ap:

    11.13 1

    x2

    1x1+x

    dx

    Ap: 1x ln(x + x2)2 + C.

    11.14 dx

    2

    x 3

    x

    Ap:

    11.15 dx

    1+x+ 3

    1+x

    Ap: 2

    1 + x 3 3

    1 + x + C.

    11.16 x+ 4x2

    3x2 dx

    Ap: