Kef�laio 1
To Aìristo Olokl rwma
1.1 Orismìc
Ac upojèsoume ìti mac dÐdetai mÐa sun�rthsh f(x), gi� thn opoÐa gnwrÐzou-me ìti èqei prokÔyei san par�gwgoc mi�c �llhc sun�rthshc σ(x), dhlad dσ(x)
dx= f(x) kaÐ ìti h f(x) eÐnai gnwst kai h σ(x) �gnwsth. ZhteÐtai na
prosdiorÐsoume thn σ(x), dojeÐshc thc f(x). 'Ena tètoio prìblhma lègetaidiaforik exÐswsh . H lÔsh thc eÐnai mi� �llh sun�rthsh, poÔ onom�zetaiaìristo olokl rwma kai sumbolÐzetai
∫f(x)dx. Genik�:
Je¸rhma 1.1 K�je sun�rthsh f(x), suneq c sto anoiktì di�sthma (a, b),eÐnai oloklhr¸simh sto (a, b).
Je¸rhma 1.2 K�je sun�rthsh, èqousa aìristo olokl rwma sto (a, b), jaèqei �peira aìrista oloklhr¸mata, ta opoÐa ja diafèroun metaxÔ touc kat�mÐa stajer� posìthta C.
Epomènwc, o prohgoÔmenoc tÔpoc genikeÔetai wc ex c:
dσ(x)
dx= f(x) ⇔ σ(x) =
∫f(x)dx + C
1.3 H Stajer� thc Olokl rwshc
H stajer� C, pou anafèrame prohgoumènwc apoteleÐ domikì kai anapìspastostoiqeÐo tou oloklhr¸matoc kai kalì eÐnai na mhn paraleÐpetai potè. 'OtandÐdontai orismènec plhroforÐec (arqikèc sunj kec), h stajer� aut mporeÐna prosdiorisjeÐ.
1
2 KEF�ALAIO 1. TO A�ORISTO OLOKL�HRWMA
1.4 Stoiqei¸dh Aìrista Oloklhr¸mata
Qrhsimopoi¸ntac ton orismì, mporoÔme na apodeÐxoume touc parak�tw tÔ-pouc twn stoiqeiwd¸n aorÐstwn oloklhrwm�twn. Pr�gmati, parag¸gish twndeutèrwn mel¸n mac dÐdei p�nta ta pr¸ta.
∫xadx =
xa+1
a + 1+ C, a ∈ R, a 6= −1 ,
∫ 1
xdx = ln |x|+ C
∫exdx = ex + C
∫ηµxdx = −συνx + C ,
∫συνxdx = ηµx + C
∫ dx
ηµ2x= −σϕx + C ,
∫ dx
συν2x= εϕx + C
∫ dx√1− x2
= τoξηµx + C , −∫ dx√
1− x2= τoξσυνx + C
∫ dx
1 + x2= τoξεϕx + C , −
∫ dx
1 + x2= τoξσϕx + C
1.5 Basikèc Idiìthtec
IsqÔoun oi parak�tw idiìthtec:
∫[f(x) + g(x)]dx =
∫f(x)dx +
∫g(x)dx
∫λ · f(x)dx = λ ·
∫f(x)dx, λ ∈ R
'Opwc blèpoume den up�rqei idiìthta pou na perigr�fei thn sumperifor� touoloklhr¸matoc sqetik� me to ginìmeno sunart sewn. Aut eÐnai kai h aitÐathc duskolÐac upologismoÔ oloklhrwm�twn, en antijèsei me thn parag¸gishpou akoloujeÐ sugkekrimèno algìrijmo.
1.6. LUM�ENES ASK�HSEIS. 3
1.6 Lumènec Ask seic.
1.1 UpologÐsate to aìristo olokl rwma:∫(4x3 − 5x2 + 6x− 1)dx.
LÔsh: Diadoqik� èqoume,∫(4x3 − 5x2 + 6x− 1)dx =
∫4x3dx−
∫5x2dx +
∫6xdx−
∫1dx =
= 4∫
x3dx−5∫
x2dx+6∫
xdx−1∫
dx = 4x3+1
3 + 1−5
x2+1
2 + 1+6
x1+1
1 + 1−1
x0+1
0 + 1+C =
= 4x4
4− 5
x3
3+ 6
x2
2− 1
x1
1+ C = x4 − 5
3x3 + 3x2 − x + C
q.e.d.
1.2 UpologÐsate to aìristo olokl rwma:∫(√
x + 13√x
)dx.
LÔsh: Diadoqik� èqoume,∫(√
x +13√
x)dx =
∫x
12 dx +
∫x−
13 dx =
=x
12+1
12
+ 1+
x−13+1
−13
+ 1+ C =
2x√
x
3+
3
23√
x2 + C
q.e.d.
1.3 ProsdiorÐsate kampÔlh, dierqomènh apì to shmeÐo M(1, 5) kaimè klÐsh 4x.
LÔsh: 'Estw y = f(x) h exÐswsh thc sugkekrimènhc kampÔlhc. AfoÔ h klÐshthc eÐnai 4x ja èqoume df
dx= 4x. Me olokl rwsh èqoume f(x) =
∫4xdx =
2x2+C. Epeid to shmeÐo M an kei sthn kampÔlh, èpetai ìti f(1) = 5, opìte5 = 2 + C, C = 3. 'Ara telik�, h zhtoumènh kampÔlh eÐnai h f(x) = 2x2 + 3.
q.e.d.
1.4 Ta oriak� èsoda mÐac epiqeÐrhshc dÐdontai apì thn sqèsh:
dR
dq= 22− 4q + 7
√q
E�n thn qronik stigm mhdèn èsoda den up�rqoun, breÐte thn su-n�rthsh esìdwn.
4 KEF�ALAIO 1. TO A�ORISTO OLOKL�HRWMA
LÔsh: Apì thn sqèsh dRdq
= 22− 4q + 7√
q èqoume ìti
R(q) =∫
(22− 4q + 7√
q)dq ⇒
⇒ R(q) = 22q − 4 · q2
2+ 7 · q3/2
3/2+ C = 22q − 2q2 +
14
3· q3/2 + C
Mac dÐdetai akìma ìti R(0) = 0, apì ìpou prokÔptei ìti C = 0 kai telik�:
R(q) = 22q − 2q2 +14
3· q3/2
q.e.d.
1.7 Ask seic Proc EpÐlush.
UpologÐsate ta parak�tw oloklhr¸mata:
1.5∫ 4x3−5x2
x2 dx.
Ap: 2x2 − 5x + C.
1.6∫(2x2 − 1
x2 + x)dx
Ap: 23x3 + 1
x+ x2
2+ C.
1.7∫(4√
x− 5x2 + 6x−12 − 1)dx
Ap: 83x
32 − 5
3x3 + 12
√x− x + C.
1.8∫( 1
x4 + 14√x− 4)dx
Ap:− 13x3 + 4
3
4√
x3 − 4x + C.
1.9∫( 3√
x− x√x
+ 7x2
3√x2−√
x)dx
Ap: 34x
43 − 2
3x
32 + 3x
73 − 2
3x
32 + C.
1.10∫(ex − xe)dx
Ap: ex − xe+1
e+1+ C.
1.11 ProsdiorÐsate kampÔlh, dierqomènh apì ta shmeÐa (1, 2), (14,−10) kaÐ
mè klÐsh antistrìfwc an�logh tou x2.
1.7. ASK�HSEIS PROS EP�ILUSH. 5
Ap: f(x) = − 4x
+ 6.
1.12 E�n h sun�rthsh oriakoÔ kìstouc eÐnai 5 − 6q2 + 7q3, breÐte thn su-n�rthsh tou kìstouc.
Ap: 5q − 2q3 + 74q4 + C
1.13 E�n h sun�rthsh oriak¸n esìdwn eÐnai 16− 4q− 1q, breÐte thn sun�r-
thsh twn mèswn esìdwn.
Ap: 1q(16q − 2q2 − ln q + C)
1.14 H oriak z thsh enìc proðìntoc dÐdetai apì thn sqèsh: dDdp
= −0.1.E�n gnwrÐzoume ìti, ìtan h tim tou proðìntoc eÐnai 3 qrhmatikèc mon�dec,zhtoÔntai 11 mon�dec proðìntoc, breÐte ta olik� èsoda ìtan h tim ja eÐnai 4.5qrhmatikèc mon�dec.
Ap: 11.3p− 0.1p2
Kef�laio 2
Olokl rwsh me thn Mèjodo twnProsdioristèwn Suntelest¸n
2.1 H Mèjodoc
H mèjodoc aut efarmìzetai sun jwc se oloklhr¸mata thc morf c:
I =∫
π(x)ϕ(x)dx
ìpou π(x) polu¸numo kai ϕ(x) ekjetik sun�rthsh. (p.q. ex, 2x, 5x2.) Jew-
roÔme ìti to olokl rwma pou zht�me, I, gr�fetai:
I = (Axk + Bxk−1 + Γxk−2 + · · ·+ E) · ϕ(x)
ìpou k ènac, kat�llhla epilegmènoc, bajmìc poluwnÔmou. AfoÔ to I eÐnaiaìristo olokl rwma, ja isqÔei profan¸c h sqèsh:
dI
dx= π(x)ϕ(x)
Antikajist¸ntac to I me to Ðso tou kai k�nontac pr�xeic, prosdiorÐzoumetouc suntelestèc A, B, Γ, . . . kai telik� to I.
2.2 Lumènec Ask seic.
2.1 UpologÐsate me thn mèjodo twn prosdioristèwn suntelest¸nto olokl rwma:
∫x3e2xdx.
7
8KEF�ALAIO 2. OLOKL�HRWSHME THNM�EJODOTWNPROSDIORIST�EWN SUNTELEST�WN
LÔsh: Upojètoume ìti to zhtoÔmeno olokl rwma I, èqei thn morf :
I = (Ax3 + Bx2 + Γx + ∆)e2x
ìpou A, B, Γ kai ∆ suntelestèc proc prosdiorismì. Apì thn sqèsh: dIdx
=x3e2x, èqoume:
(Ax3 + Bx2 + Γx + ∆)′e2x + (Ax3 + Bx2 + Γx + ∆)(e2x)′ = x3e2x
�ra,
(3Ax2 + 2Bx + Γ)e2x + 2(Ax3 + Bx2 + Γx + ∆)′e2x = x3e2x ⇒
⇒ (2A)x3e2x + (3A + 2B)x2e2x + (2B + 2Γ)xe2x + (Γ + 2∆)e2x = x3e2x
Exis¸nontac touc suntelestèc twn dÔo mer¸n thc exÐswshc, paÐrnoume tosÔsthma:
2A = 1
3A + 2B = 0
2B + 2Γ = 0
Γ + 2∆ = 0
ap' ìpou brÐskoume:
A =1
2, B = −3
4, Γ =
3
4, ∆ = −3
8
kai �ra to zhtoÔmeno olokl rwma eÐnai:
I = (1
2x3 − 3
4x2 +
3
4x− 3
8)e2x + C
q.e.d.
2.2 Me th mèjodo twn prosdioristèwn suntelest¸n upologÐsateto olokl rwma: I =
∫5x22xdx.
LÔsh: JewroÔme ìti: I = (Ax2 + Bx + Γ)2x. Apì thn sqèsh dIdx
= 5x22x,èqoume:
(Ax2 + Bx + Γ)′2x + (Ax2 + Bx + Γ)(2x)′ = 5x22x ⇒
(2Ax + B)2x + (Ax2 + Bx + Γ)2x ln 2 = 5x22x
2.2. LUM�ENES ASK�HSEIS. 9
Exis¸nontac touc suntelestèc paÐrnoume:
A ln 2 = 5
2A + B ln 2 = 0
B + Γ ln 2 = 0
ap' ìpou prokÔptei ìti:
A =5
ln 2, B = − 10
(ln 2)2, Γ =
10
(ln 2)3
kai �ra to zhtoÔmeno olokl rwma eÐnai:
I =
(5
ln 2x2 − 10
(ln 2)2x +
10
(ln 2)3
)2x + C
q.e.d.
2.3 UpologÐsate to olokl rwma I =∫
x2αx2dx.
LÔsh: JewroÔme ìti to zhtoÔmeno olokl rwma I, èqei thn morf : I =(Ax + B)αx2
. H sqèsh dIdx
= x2αx2, ja d¸sei:
(Ax + B)′αx2
+ (Ax + B)(αx2
)′ = x2αx2
opìteAαx2
+ (Ax + B)2xαx2
ln α = x2αx2
Exis¸nontac touc suntelestèc èqoume:
2A ln α = 1
A + 2B ln α = 0
ap' ìpou brÐskoume ìti:
A =1
2 ln α, B = − 1
4(ln α)2
�ra to zhtoÔmeno olokl rwma eÐnai:
I =
(1
2 ln αx− 1
4(ln α)2
)αx2
+ C
q.e.d.
10KEF�ALAIO 2. OLOKL�HRWSHME THNM�EJODOTWNPROSDIORIST�EWN SUNTELEST�WN
2.4 UpologÐsate to olokl rwma: I =∫(x + 1)1000(5x + 7)dx.
LÔsh: Upojètoume ìti to I èqei thn morf I = (x + 1)1001(Ax + B). Hsqèsh dI
dx= (5x + 7)(x + 1)1000 ja mac d¸sei:
(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 aplopoi¸ntac to (x + 1)1000, paÐrnoume:
Ax + A + 1001Ax + 1001B = 5x + 7
ap' ìpou exis¸nontac touc suntelestèc, prokÔptei:
A =5
1002, B =
7009
1003002
'Ara, to zhtoÔmeno olokl rwma eÐnai:
I =(
5
1002x +
7009
1003002
)(x + 1)1001 + C
q.e.d.
2.3 Ask seic Proc EpÐlush
2.5 UpologÐsate to olokl rwma∫
6x6exdx.
Ap: (4320 − 4320x + 2160x2 − 720x3 + 180x4 −36x5 + 6x6)ex + C
2.6 UpologÐsate to olokl rwma∫
11x2e2x+7dx.
Ap: e2x+7(114− 11
2x + 11
2x2) + C
2.7 UpologÐsate to olokl rwma∫
6x3αxdx.
Ap: 6αx
ln a(−3
2+ 2x− 3
2x2 + x3) + C
2.8 UpologÐsate to olokl rwma∫
x25x3dx.
Ap: 13 ln 5
5x3
2.9 UpologÐsate to olokl rwma∫(2x + 8)999(7x + 9)dx.
Ap:(
72002
x + 89622002000
)(2x + 8)1000
Kef�laio 3
Olokl rwsh me Antikat�stash
H olokl rwsh me antikat�stash basÐzetai kurÐwc ston parak�tw metasqh-matismì:
∫f(g(x))g′(x)dx =
∫f(ω)dω oπoυ ω = g(x)
Parat rhsh 3.1 H olokl rwsh me antikat�stash efarmìzetai sun jwc sesunart seic pou èna tm ma touc eÐnai sÔnjeto, thc morf c f(g(x)) kai to �llo¨perièqei¨ thn par�gwgo thc g(x), mporoÔme eÔkola na thn sqhmatÐsoume.
3.1 Lumènec Ask seic
3.1 UpologÐsate to aìristo olokl rwma:∫ dx
2x+3.
LÔsh: Jètoume ω = 2x + 3, opìte dωdx
= 2, dω = 2dx kai to olokl rwmagÐnetai
∫ dx2x+3
=∫ dω
2ω= 1
2
∫ dωω
= 12ln |ω|+ C = 1
2ln |2x + 3|+ C.
q.e.d.
3.2 UpologÐsate to aìristo olokl rwma:∫ dx
4−5x.
LÔsh: Jètoume ω = 4 − 5x, opìte dωdx
= −5, dx = dω−5
kai to olokl rwma
gÐnetai∫ dx
4−5x=∫ dω
−5ω= −1
5
∫ dωω
= −15ln |ω|+ C = −1
5ln |4− 5x|+ C.
q.e.d.
3.3 UpologÐsate to aìristo olokl rwma:∫ dx
(3−2x)3.
11
12 KEF�ALAIO 3. OLOKL�HRWSH ME ANTIKAT�ASTASH
LÔsh: Jètoume ω = 3 − 2x, opìte dωdx
= −2, dx = dω−2
kai to olokl rwma
gÐnetai∫ dx
(3−2x)3=∫ dω
−2ω3 = −12
∫ dωω3 = −1
2· ω−3+1
−3+1+ C = −1
2· 1−2ω2 + C
= 14(3−2x)2
+ C.
q.e.d.
3.4 UpologÐsate to aìristo olokl rwma:∫ xdx
x2+1.
LÔsh: Jètoume ω = x2 + 1 kai èqoume dωdx
= 2x, opìte dx = dω2x
kai toolokl rwma diadoqik� gÐnetai:
∫ xdx
x2 + 1=∫ xdω
2x
ω=
1
2
∫ dω
ω=
1
2ln |ω|+ C =
1
2ln(x2 + 1) + C
q.e.d.
3.5 UpologÐsate to aìristo olokl rwma:∫(1− 2x)100dx.
LÔsh: Jètoume ω = 1 − 2x kai èqoume dωdx
= −2, opìte dx = dω−2
kai toolokl rwma diadoqik� gÐnetai:
∫(1− 2x)100dx =
∫ω100 dω
−2= −1
2
ω101
101+ C = − 1
202(1− 2x)101 + C
q.e.d.
3.6 UpologÐsate to aìristo olokl rwma:∫
ex3+2x2dx.
LÔsh: Jètoume ω = x3 + 2, opìte dωdx
= 3x2, dx = dω3x2 kai to olokl rwma
diadoqik� gÐnetai:∫ex3+2x2dx =
∫eωx2 dω
3x2=
1
3
∫eωdω =
1
3eω + C =
1
3ex3+2 + C
q.e.d.
3.7 UpologÐsate to aìristo olokl rwma:∫
3xdx.
LÔsh: Jètontac ω = 3x, èqoume dωdx
= 3x ln 3, dx = dω3x ln 3
kai to olokl rwmadiadoqik� gÐnetai:∫
3xdx =∫
3x dω
3x ln 3=
1
ln 3
∫dω =
1
ln 3ω + C =
1
ln 33x + C
q.e.d.
3.1. LUM�ENES ASK�HSEIS 13
3.8 UpologÐsate to aìristo olokl rwma:∫
εϕxdx.
LÔsh: Kat' arq�c gr�foume εϕx = ηµxσυνx
kai met� jètoume ω = συνx, opìtedωdx
= −ηµx, dx = − dωηµx
kai to olokl rwma diadoqik� gÐnetai:
∫εϕxdx =
∫ ηµx
συνxdx =
∫− ηµx
συνx· dω
ηµx= −
∫ dω
ω= − ln |ω|+C = − ln |συνx|+C
q.e.d.
3.9 UpologÐsate to aìristo olokl rwma:∫ 1
συν2xdx.
LÔsh: Jètontac ω = εϕx, èqoume dωdx
= 1συν2x
, dx = συν2xdω kai to olo-kl rwma diadoqik� gÐnetai:
∫ 1
συν2xdx =
∫ συν2xdω
συν2x=∫
dω = ω + C = εϕx + C
q.e.d.
3.10 UpologÐsate to aìristo olokl rwma:∫ συν3xdx
ηµ4x.
LÔsh: Jètontac ω = ηµx, èqoume dωdx
= συνx, dx = dωσυνx
kai to olokl rwmadiadoqik� gÐnetai:
∫ συν3xdx
ηµ4x=∫ συν2xσυνxdx
ηµ4x=∫ (1− ηµ2x)συνxdx
ηµ4x=
=∫ (1− ω2)συνx dω
συνx
ω4=∫ dω
ω4−∫ dω
ω2= − 1
3ω3+
1
ω+C = − 1
3ηµ3x+
1
ηµx+C
q.e.d.
3.11 UpologÐsate to aìristo olokl rwma:∫ xσυνxdx
xηµx+συνx.
LÔsh: Jètontac ω = x ·ηµx+συνx, èqoume dωdx
= x ·συνx, x ·συνxdx = dωkai to olokl rwma diadoqik� gÐnetai:∫ xσυνxdx
xηµx + συνx=∫ dω
ω= ln |ω|+ C = ln |xηµx + συνx|+ C
q.e.d.
3.12 UpologÐsate to aìristo olokl rwma:∫ dx
x(ln x+3).
14 KEF�ALAIO 3. OLOKL�HRWSH ME ANTIKAT�ASTASH
LÔsh: Jètontac ω = ln x + 3, èqoume dωdx
= 1x, dω = dx
xkai to olokl rwma
diadoqik� gÐnetai:∫ dx
x(ln x + 3)=∫ dω
ω= ln |ω|+ C = ln | ln x + 3|+ C
q.e.d.
3.13 UpologÐsate to aìristo olokl rwma:∫ x2dx
4√x3+2.
LÔsh: Jètontac ω = x3 + 2, èqoume dωdx
= 3x2, dω = 3x2dx kai to olokl -rwma diadoqik� gÐnetai:
∫ x2dx4√
x3 + 2=
1
3
∫ dω4√
ω=
1
3· ω− 3
4(−3
4
) + C = +4
9(x3 + 2)
34 + C
q.e.d.
3.14 UpologÐsate to aìristo olokl rwma:∫ √
x2 − 2x4dx me 0 <x < 1√
2.
LÔsh: Jètontac ω = 1 − 2x2, èqoume dωdx
= −4x, dω = −4xdx kai toolokl rwma diadoqik� gÐnetai:∫ √
x2 − 2x4dx =∫ √
1− 2x2xdx = −1
4
∫ √ωdω =
= −1
4· ω3/2
3/2= −1
4· (1− 2x2)3/2
3/2+ C
q.e.d.
3.2 Ask seic Proc EpÐlush.
UpologÐsate ta parak�tw aìrista oloklhr¸mata:
3.15∫ 1
ax+bdx
Ap: 1aln |ax + b|+ C
3.16∫ 1
(ax+b)n dx
Ap: 1a(1−n)(b+ax)n−1
3.2. ASK�HSEIS PROS EP�ILUSH. 15
3.17∫ xdx
(x2+5)5.
Ap: − 18(5+x2)4
+ C
3.18∫ xdx
(x2+4)3.
Ap: − 14(4+x2)2
+ C
3.19∫(5 + 2x)1000dx.
Ap: (5+2x)1001
2002+ C
3.20∫(ax + b)ndx
Ap: (b+ax)n+1
a(n+1)+ C
3.21∫
x3(x4 + 2)34 dx
Ap: 17(2 + x4)7/4 + C
3.22∫(1− x3)2x2dx
Ap: x3
3− x6
3+ x9
9+ C
3.23∫(2x2 + 3)
13 xdx
Ap: 12(9
8+ 3x2
4)(3 + 2x2)1/3 + C
3.24∫(x2 − x)4(2x− 1)dx
Ap: −x5
5+x6−2x7+2x8−x9+ x10
5+C
3.25∫
axdx
Ap: ax
ln a+ C
3.26∫
ηµ(ax + b)dx
Ap: −συν(ax+b)a
+ C
3.27∫
xηµ(3 + x2)dx
Ap: −συν(3+x2)2
+ C
3.28∫ 1
ηµ2xdx
16 KEF�ALAIO 3. OLOKL�HRWSH ME ANTIKAT�ASTASH
Ap: −σϕx + C
3.29∫
ηµx5συνxdx
Ap: −5συνx
ln 5+ C.
3.30∫
σϕxdx
Ap: ln(ηµx) + C
3.31∫ ηµ2xdx√
1+ηµ2x
Ap: 2√
1 + ηµ2x + C.
3.32∫ τoξηµxdx√
1−x2
Ap: 12τoξηµ2x.
3.33∫
eax+bdx
Ap: 1aeax+b + C.
3.34∫
e√
x dx2√
x
Ap: e√
x + C.
3.35∫ e2x−7ex+2
ex dx
Ap: −2e−x + ex − 7x + C.
3.36∫ e
1x dxx2
Ap: −e1x + C.
3.37∫ e
1x2 dxx3
Ap: −12e
1x2 + C.
3.38∫ 3 ln x+2
xdx
Ap: 2 ln x + 32ln2 x + C.
3.39∫ √
x5 + 3 · x4dx
3.2. ASK�HSEIS PROS EP�ILUSH. 17
Ap: 15(2 + 2x5
3)√
3 + x5 + C.
3.40∫ dx√
4−x2
Ap: τoξηµ(x/2) + C.
3.41∫ (x+1)dx√
x2+2x−4
Ap:√
x2 + 2x− 4 + C.
3.42∫
3√
1− x2 · x
Ap: 12(1− x2)1/3(−3
4+ 3x2
4) + C.
Kef�laio 4
Oloklhr¸mata mèsw thc τoξεϕx
'Otan èqoume proc olokl rwsh kl�smata, me �jroisma tetrag¸nwn stonparanomast , tìte sun jwc qrhsimopoioÔme ton tÔpo:∫ du
u2 + δ2=
1
δτoξεϕ
u
δ+ C
To u, ston parap�nw tÔpo, mporeÐ na eÐnai mÐa apl metablht kai olìklhrhsun�rthsh.
4.1 Lumènec Ask seic
4.1 UpologÐsate to olokl rwma:∫ dx
x2+4.
LÔsh: Diadoqik� èqoume:∫ dx
x2 + 4=∫ dx
x2 + 22=
1
2τoξεϕ
x
2+ C
q.e.d.
4.2 UpologÐsate to olokl rwma:∫ dx
x2+6x+13.
LÔsh: Diadoqik� èqoume∫ dx
x2 + 6x + 13=∫ dx
x2 + 6x + 9 + 4=∫ dx
(x + 3)2 + 22=
=∫ d(x + 3)
(x + 3)2 + 22=
1
2τoξεϕ
(x + 3
2
)+ C
q.e.d.
19
20 KEF�ALAIO 4. OLOKLHR�WMATA M�ESW THS τOξεϕX
4.3 UpologÐsate to olokl rwma:∫ dx
3x2+4x+11.
LÔsh: O paranomast c gÐnetai: 3x2 +4x+11 = (√
3x)2 +2 · 12·4 ·
√3 · 1√
3·
x+11 = (√
3x)2+2·√
3· 2√3·x+11 = (
√3x)2+2·
√3· 2√
3·x+( 2√
3)2−( 2√
3)2+11 =
(√
3x + 2√3)2 + (
√293)2 kai epomènwc to olokl rwma gr�fetai:
I =∫ dx
(√
3x + 2√3)2 + (
√293)2
Jètontac u =√
3x+ 2√1, èqoume du =
√3dx ⇒ dx = du√
3kai antikajist¸ntac
paÐrnoume:
I =∫ du√
3
u2 + (√
293)2
=2√1· 1√
293
· τoξεϕu√293
+ C =
=1√29
τoξεϕ
√3x + 2√3√
293
+ C =
√29
29τoξεϕ
(3x + 2√
29
)+ C
q.e.d.
4.4 UpologÐsate to olokl rwma:∫ dx
x ln2 x+10x.
LÔsh: Kat' arq�c jètoume w = ln x, opìte dwdx
= 1x⇒ dx = xdw.
Antikajist¸ntac paÐrnoume gia to olokl rwma:
∫ dx
x(lnx)2 + 10x=∫ xdw
x(w2 + 10)=∫ dw
w2 + (√
10)2=
=1√10
τoξεϕ
(w√10
)+ C =
√10
10τoξεϕ
(ln x√
10
)+ C
q.e.d.
4.5 UpologÐsate to olokl rwmv:∫ ηµx·συνx·dx
ηµ4x+συν4x.
LÔsh: DiairoÔme arijmht kai paranomast me to συν4x kai to olokl -rwma gÐnetai: ∫ ηµx · συνx · dx
ηµ4x + συν4x=∫ εϕx · 1
συν2x
(εϕ2x)2 + 1· dx
4.2. ASK�HSEIS PROS EP�ILUSH. 21
Jètoume w = εϕ2x kai èqoume dwdx
= 2εϕx · 1συν2x
, opìte dw2
= εϕx · 1συν2x
· dxkai to olokl rwma gÐnetai:
I =∫ 1
2· dw
w2 + 1=
1
2
∫ dw
w2 + 1=
1
2τoξεϕw + C =
=1
2τoξεϕ(εϕ2x) + C
q.e.d.
4.2 Ask seic Proc EpÐlush.
UpologÐsate ta oloklhr¸mata:
4.6∫ dx
2ηµ2x+5συν2x
Ap: 13τoξεϕ(
√23· εϕx) + C.
4.7∫ συνxdx
ηµ2x+7
Ap: 6√7τoξεϕ(
√17· ηµx) + C.
4.8∫ xdx
x4+9
Ap: 14τoξεϕ(x2
3) + C.
4.9∫ exdx
e2x+1
Ap: τoξεϕ(ex) + C.
4.10∫ συνxdx
(ln8(ηµx)+4)ηµx
Ap: 12τoξεϕ( ln(ηµx)
2) + C.
4.11∫ ηµxdx
9συν2x+4
Ap: 16τoξεϕ(3
2· συνx) + C.
4.12∫ dx
5x2+5x+3
Ap: − 2√21
τoξεϕ(7+10x√29
) + C.
4.13∫ dx
x2−x+1
Ap: 2√39
τoξεϕ(2x−1√3
) + C.
4.14∫ dx
113
x6−x+ 87
Ap: 2√
2123
τoξεϕ(√
769
(22x− 3)) + C.
Kef�laio 5
Paragontik Olokl rwsh
5.1 O Basikìc TÔpoc.
An kai den up�rqei genikìc tÔpoc gi� thn olokl rwsh ginomènou, h parak�twsqèsh mporeÐ na aplopoi sei poll� oloklhr¸mata:∫
f(x)g′(x)dx = f(x)g(x)−∫
f ′(x)g(x)dx
Onom�zetai paragontik olokl rwsh kai prokÔptei sunoptik� wc ex c: 'Estw-san dÔo sunart seic f(x) kai g(x). IsqÔei ìti [f(x) · g(x)]′ = f ′(x) · g(x) +f(x) · g′(x), oloklhr¸nontac èqoume
∫[f(x) · g(x)]′dx =
∫f ′(x) · g(x)dx +∫
f(x) · g′(x)dx. Epeid de,∫[f(x) · g(x)]′dx = f(x) · g(x), antikajist¸ntac
èqoume to zhtoÔmeno.
5.2 EmpeirikoÐ Kanìnec
Den up�rqei akrib c diadikasÐa gia to poi� sun�rthsh ja sunduasjeÐ methn par�gwgo. MolotaÔta, oi akìloujoi empeirikoÐ kanìnec eÐnai exairetik�qr simoi:
• 'Otan èqoume dÔnamh tou x kai trigwnometrik sun�rthsh, sundu�zoumeme thn par�gwgo, thn trigwnometrik sun�rthsh.
• 'Otan èqoume dÔnamh tou x kai ekjetik sun�rthsh, sundu�zoume methn par�gwgo, thn ekjetik sun�rthsh.
• 'Otan èqoume dÔnamh tou x kai logarijmik sun�rthsh, sundu�zoume methn par�gwgo, thn dÔnamh tou x.
• 'Otan èqoume ekjetik sun�rthsh kai trigwnometrik sun�rthsh, k�nou-me dÔo paragontikèc oloklhr¸seic me ton Ðdio sunduasmì kai met� lÔnoumeexÐswsh.
23
24 KEF�ALAIO 5. PARAGONTIK�H OLOKL�HRWSH
• EnÐote kai aplì olokl rwma mporeÐ na metatrapeÐ se paragontikì pol-laplasiazìmeno me thn posìthta x′.
5.3 Lumènec Ask seic
5.1 UpologÐsate to olokl rwma:∫
xσυνxdx.
LÔsh: Diadoqik� èqoume:∫x(ηµx)′dx = xηµx−
∫(x)′ηµxdx =
= xηµx−∫
ηµxdx = xηµx + συνx + C
q.e.d.
5.2 UpologÐsate to olokl rwma:∫
x2exdx.
LÔsh: Diadoqik� èqoume:∫x2exdx =
∫x2(ex)′dx = x2ex −
∫(x2)′exdx =
= x2ex −∫
2xexdx = x2ex − 2∫
x(ex)′dx =
= x2ex − 2xex + 2∫
(x)′exdx =
= x2ex − 2xex + 2∫
exdx = x2ex − 2xex + 2ex + C =
= ex(x2 − 2x + 2) + C
q.e.d.
5.3 UpologÐsate to olokl rwma:∫
x2ηµxdx.
LÔsh: Diadoqik� èqoume:∫x2ηµxdx =
∫x2(−συνx)′dx = −x2συνx +
∫(x2)′συνxdx =
5.3. LUM�ENES ASK�HSEIS 25
= −x2συνx + 2∫
xσυνxdx = −x2συνx + 2∫
x(ηµx)′dx =
= −x2συνx + 2xηµx− 2∫
(x)′ηµxdx =
= −x2συνx + 2xηµx− 2∫
ηµxdx = −x2συνx + 2xηµx + 2συνx + C
q.e.d.
5.4 UpologÐsate to olokl rwma:∫
x3 ln xdx.
LÔsh: Diadoqik� èqoume, efarmìzontac paragontik :∫x3 ln xdx =
∫ (x4
4
)′
ln xdx =x4
4ln x−
∫ x4
4· 1
xdx =
=x4
4ln x− 1
4
∫x3dx =
x4
4ln x− 1
4· x4
4+ C =
=x4
4ln x− x4
16+ C
q.e.d.
5.5 UpologÐsate to olokl rwma:∫
e−xηµxdx
LÔsh: OrÐzoume w =∫
e−xηµxdx kai efarmìzontac paragontik olokl rw-sh, èqoume diadoqik�:
I =∫
(−e−x)′ηµxdx = −e−xηµx−∫
(−e−x(ηµx)′)dx
= −e−xηµx +∫
(e−xσυνx)dx = −e−xηµx + I0
ìpou I0 =∫
e−xσυνxdx. To nèo autì olokl rwma upologÐzetai me mÐa akìmaparagontik olokl rwsh kai èqoume:
I0 =∫
(−e−x)′συνxdx = −e−xσυνx−∫
(−e−x)(συνx)′dx
= −e−xσυνx−∫
e−xηµxdx = −e−xσυνx− I ⇒ I0 = −e−xσυνx− I
�ra telik� I = −e−xηµx − e−xσυνx − I. LÔnontac algebrik� wc proc I,paÐrnoume:
I =1
2(−e−xηµx− e−xσυνx) + C
q.e.d.
26 KEF�ALAIO 5. PARAGONTIK�H OLOKL�HRWSH
5.6 UpologÐsate to olokl rwma:∫
ηµ(ln x)dx
LÔsh: Jètoume w = ln x kai �ra x = ew ⇒ dx = ewdw. To olokl rwmagÐnetai:
I =∫
ηµwewdw =∫
ηµw(ew)′dw =
= ηµwew −∫
(ηµw)′ewdw = ηµwew −∫
συνwewdw = ηµwew − Io
UpologÐzoume t¸ra to olokl rwma Io, qrhsimopoi¸ntac p�li paragontik :
Io =∫
συνwewdw =∫
συνw(ew)′dw =
= συνwew −∫
(συνw)′ewdw =
= συνwew +∫
ηµwewdw = συνwew + I
Antikajist¸ntac thn tim tou Io sthn sqèsh gia to I, èqoume:
I = ηµwew − συνwew − I ⇒ 2I = ηµwew − συνwew ⇒
⇒ I =1
2· (ηµwew − συνwew) + C
Me antÐstrofh antikat�stash paÐrnoume telik�:
I =x
2[ηµ(ln x)− συν(ln x)] + C
q.e.d.
5.7 UpologÐsate to olokl rwma:∫
ηµx · ηµ(3x)dx
LÔsh: OrÐzoume
I =∫
ηµx · ηµ(3x)dx =∫
(−συνx)′ηµ(3x)dx =
= −συνxηµ(3x)−∫
[−συνx(ηµ(3x))′]dx =
= −συνxηµ(3x) + 3∫
συνxσυν(3x)dx = −συνxηµ(3x) + 3Io
5.3. LUM�ENES ASK�HSEIS 27
UpologÐzoume t¸ra to Io, p�li me paragontik :
Io =∫
συνxσυν(3x)dx =∫
(ηµx)′συν(3x)dx =
= ηµxσυν(3x)−∫
ηµx(συν(3x))′dx =
= ηµxσυν(3x) + 3∫
ηµxηµ(3x)dx = ηµxσυν(3x) + 3I
Antikajist¸ntac kai lÔnontac wc proc I, èqoume
I = −συνxηµ(3x)+3(ηµxσυν(3x)+3I) ⇒ I =1
8(συνxηµ(3x)−3ηµxσυν(3x))+C
q.e.d.
5.8 UpologÐsate to olokl rwma:∫
xηµ(ax + b)dx.
LÔsh: OrÐzoume I =∫
xηµ(ax + b)dx kai èqoume diadoqik�:
I =(−1
a
) ∫x(συν(ax+b))′dx =
(−1
a
)xσυν(ax+b)+
(1
a
) ∫(x)′συν(ax+b)dx =
=(−1
a
)xσυν(ax + b) +
(1
a
) ∫συν(ax + b)dx =
=(−1
a
)xσυν(ax + b) +
(1
a2
)ηµ(ax + b) + C
q.e.d.
5.9 UpologÐsate to Olokl rwma∫ xex
(x+1)2dx.
LÔsh: Zht�me arijmoÔc A kai B ètsi ¸ste:
x
(x + 1)2=
A
x + 1+
B
(x + 1)2=
=x
(x + 1)2=
Ax + A + B
(x + 1)2
opìte A = 1 kai A + B = 0, �ra A = 1 kai B = −1. To olokl rwma t¸ragÐnetai: ∫ xex
(x + 1)2dx =
∫ ex
x + 1dx−
∫ ex
(x + 1)2dx
28 KEF�ALAIO 5. PARAGONTIK�H OLOKL�HRWSH
UpologÐzoume pr¸ta to pr¸to olokl rwma me paragontik :∫ ex
x + 1dx =
∫ (ex)′
x + 1dx =
=ex
x + 1−∫
ex · ( 1
x + 1)′dx =
ex
x + 1+∫ ex
(x + 1)2dx
�ra telik�∫ xex
(x + 1)2dx =
ex
x + 1+∫ ex
(x + 1)2dx−
∫ ex
(x + 1)2dx =
ex
x + 1+ C
q.e.d.
5.10 UpologÐsate tautìqrona ta oloklhr¸mata: I1 =∫
eaxσυν(bx)dxkai I2 =
∫eaxηµ(bx)dx.
LÔsh: UpologÐzoume pr¸ta to I1 paragontik�:
I1 =∫
(1
aeax)′συν(bx)dx =
1
aeaxσυν(bx)− 1
a
∫eax(συν(bx))′dx =
=1
aeaxσυν(bx)− 1
a
∫eaxb(−ηµ(bx))dx
⇒ I1 =1
aeaxσυν(bx) +
b
aI2 (5.1)
OmoÐwc gia to I2 èqoume:
I2 =∫
(1
aeax)′ηµ(bx)dx =
=1
aeaxηµ(bx)− 1
a
∫eax(ηµ(bx))′dx =
=1
aeaxηµ(bx)− 1
a
∫eaxb · συν(bx)dx
⇒ I2 =1
aeaxηµ(bx)− b
aI1 (5.2)
EpilÔontac to sÔsmhma twn exis¸sewn (5.1) kai (5.2), wc prìc I1 kai I2,èqoume telik�:
I1 =1
(a2 + b2)[aeaxσυν(bx) + beaxηµ(bx)] + C
I2 =1
(a2 + b2)[aeaxηµ(bx) + beaxσυν(bx)] + C
q.e.d.
5.3. LUM�ENES ASK�HSEIS 29
5.11 UpologÐsate to olokl rwma:∫ x
συν2xdx.
LÔsh: Blèpoume ìti (εϕx)′ = 1συν2x
kai epomènwc ja doulèyoume parago-ntik�. Diadoqik� èqoume:∫ x
συν2xdx =
∫x(εϕx)′dx = xεϕx−
∫(x)′εϕxdx =
= xεϕx−∫
εϕxdx
Xèroume ìti∫
εϕxdx = − ln(|συνx|) + C kai �ra telik�∫ x
συν2xdx = xεϕx + ln(|συνx|) + C
q.e.d.
5.12 Qrhsimopoi¸ntac paragontik olokl rwsh upologÐsate toolokl rwma
∫εϕ2xdx.
LÔsh: OrÐzoume I =∫
εϕ2xdx kai diadoqik� èqoume:
I =∫ ηµ2x
συν2xdx =
∫ ηµx · ηµx
συν2xdx =
∫ ηµx · (−συνx)′
συν2xdx =
= −ηµx · συνx
συν2x+∫ (
ηµx
συν2x
)′· συνxdx =
= −εϕx +∫ (ηµx)′συν2x− ηµx(συν2x)′
συν4x· συνxdx =
= −εϕx +∫ συν3x + ηµ2x · 2 · συνx
συν3xdx =
= −εϕx +∫
(1 + 2εϕ2x)dx
'Ara èqoume:
I = −εϕx +∫
dx + 2∫
εϕ2xdx = −εϕx + x + 2I
kai lÔnontac wc prìc I, paÐrnoume telik�:
I = εϕx− x + C
q.e.d.
30 KEF�ALAIO 5. PARAGONTIK�H OLOKL�HRWSH
5.13 UpologÐsate to olokl rwma∫π(x)συν(βx)dx, β 6= 0
ìpou π(x) polu¸numo µ-bajmoÔ.
LÔsh: Oloklhr¸nontac paragontik� èqoume:∫π(x)συν(βx)dx =
∫π(x)
(1
βηµ(βx)
)′
dx =
=1
βπ(x)ηµ(βx)− 1
β
∫π′(x)ηµ(βx)dx =
=1
βπ(x)ηµ(βx) +
1
β
∫π′(x)συν
(βx +
π
2
)dx
'Ara deÐxame ìti: ∫π(0)(x)συν
(βx + 0 · π
2
)dx =
=1
β· π(0)(x)ηµ
(βx + 0 · π
2
)+
1
β
∫π′(x)συν
(βx + 1 · π
2
)dx
ìpou π(0)(x) = π(x) h mhdenik par�gwgoc tou π(x). Epanalamb�nontac ta
Ðdia gia to olokl rwma∫
π′(x)συν(βx + π
2
)dx, èqoume:∫
π′(x)συν(βx + 1 · π
2
)dx =
1
βπ′(x)ηµ
(βx + 1 · π
2
)+∫
π′′(x)συν(βx + 2 · π
2
)dx
�ra genik� èqoume: ∫π(k)(x)συν
(βx + k · π
2
)dx =
=1
βπ(k)(x)ηµ
(βx + k · π
2
)+∫
π(k+1)(x)συν(βx + (k + 1) · π
2
)dx
UpologÐzontac ìla ta oloklhr¸mata gia k = 0, 1, 6, . . . , µ, lamb�nontac up�ìyhn ìti π(µ+1)(x) = 0 kai antikajist¸ntac antÐstrofa, èqoume telik�:∫
π(x)συν(βx)dx =µ∑
λ=0
1
βλ+1· π(λ)(x)ηµ
(βx + λ · π
2
)
q.e.d.
5.4. ASK�HSEIS PROS EP�ILUSHN 31
5.14 Se èna arqikì kef�laio 500 qrhmatik¸n mon�dwn, gÐnontai
ependÔseic b�sei tou tÔpou I(t) = e0.2√
t. BreÐte to kef�laio thnqronik stigm t = 3.
LÔsh: IsqÔei h basik sqèsh:
dK
dt= I(t) ⇒ dK
dt= e0.2
√t ⇒ K(t) =
∫e0.2
√tdt
Ja oloklhr¸soume me antikat�stash. Jètoume ω = 0.2√
t kai èqoume
dω = 0.2 · 1
2· t−1/2dt =
0.1√tdt ⇒
⇒ dt =
√t
0.1dω =
ω
0.02dω ⇒
⇒∫
e0.2√
tdt =1
0.02
∫eωdω = 50
∫eωωdω
Qrhsimopoi¸ntac t¸ra paragontik olokl rwsh èqoume:∫eωωdω =
∫(eω)′ωdω = ωeω −
∫eω(ω)′dω =
ωeω −∫
eωdω = ωeω − eω + C
'Ara
K(t) = 50(0.2√
te0.2√
t − e0.2√
t + C)
Gi� t = 0, K(6) = 500 kai epomènwc 500 = 50(0 − 1 + C) ⇒ C = 11 kaitelik�
K(t) = 50(0.2√
te0.2√
t − e0.2√
t + 11)
kai me antikat�stash brÐskoume K(3) = 503.792.
q.e.d.
5.4 Ask seic proc EpÐlushn
UpologÐsate ta oloklhr¸mata
5.15∫
xηµxdx
Ap: −xσυνx + ηµx + C
32 KEF�ALAIO 5. PARAGONTIK�H OLOKL�HRWSH
5.16∫
τoξηµxdx
Ap: xτoξηµx +√
1− x2 + C
5.17∫
xτoξεϕxdx
Ap: 12(x2 + 1)τoξεϕx− x
2+ C
5.18∫ dx
ηµ3x
Ap: − σϕx2ηµx
+ 12ln∣∣∣ 1ηµx
− σϕx∣∣∣+ C
5.19∫
x ln(x2 + 7)dx
Ap: −x2
2+ 1
2(x2 + 7) ln(x2 + 7) + C
5.20∫
ln(εϕx) 9συν2x
dx
Ap: −9εϕx + 9 ln(εϕx)εϕx + C
5.21∫ 1
xln xdx
Ap: 12(ln x)2 + C
5.22∫
x√
x + 1dx
Ap: 215
(√
x + 1)(3x2 + x− 2)
5.23∫(x + 3)(x + 2)−4dx
Ap: − 13(2+x)3
− 12(2+x)2
5.24∫
3xηµxdx
Ap: 3x(ln x·ηµx−συνx)
ln2 3+1
5.25∫
xτoξηµ(
ax
)dx
Ap: x2
2τoξηµ
(ax
)+ ax
2
√x2−a2
x2 + C
5.26∫
x3(ln x)2dx
Ap: x4
4ln2 x− x4
8ln x + x4
32+ C
5.4. ASK�HSEIS PROS EP�ILUSHN 33
5.27∫
ln(√
x− 1)dx
Ap: ln(√
x− 1)(x− 1) + 1−x2
+ C
5.28∫
x22xdx
Ap: 2x(
2ln3 2
− 2xln2 2
+ x2
ln 2
)5.29 Na apodeiqjeÐ o tÔpoc:∫
f(x)dx = xf(x)−∫
xf ′(x)dx
5.30 Na apodeiqjeÐ o tÔpoc:∫f(x)g′′(x)dx + f ′(x)g(x) =
∫g(x)f ′′(x)dx + f(x)g′(x)
5.31 UpologÐsate to olokl rwma:∫ln[σ(t)]e−rtdt
ìpou r ∈ R kai σ(t) sun�rthsh tou t.
5.32 UpologÐsate to olokl rwma:∫U(C(t))e−rtdt
ìpou r ∈ R kai σ(t) sun�rthsh tou t.
5.33 UpologÐsate to olokl rwma:
T = k∫
u2(A− u)γ−1du
ìpou A ∈ R kai γ > 0.
5.34 DeÐxate ìti, e�n to p(x) eÐnai polu¸numo n-bjmoÔ, tìte:∫exp(x) = ex[p(x)− p′(x) + p′′(x)− · · ·+ (−1)np(n)(x)]
Kef�laio 6
AnagwgikoÐ TÔpoi
6.1 Genik�
Me thn bo jeia thc paragontik c olokl rwshc mporoÔme na apodeÐxoumetÔpouc anagwg c enìc oloklhr¸matoc se �llo aploÔstero. AutoÐ oi tÔpoionom�zontai anagwgikoÐ.
6.2 TrigwnometrikoÐ AnagwgikoÐ TÔpoi
Oi k�twji tÔpoi eÐnai qrhsimìtatoi:E�n Iν =
∫ηµνxdx, tìte:
Iν = −ηµν−1xσυνx
ν+
ν − 1
νIν−2
E�n Iν =∫
συννxdx, tìte:
Iν =συνν−1xηµx
ν+
ν − 1
νIν−2
6.3 Anagwgikìc TÔpoc Rht c Olokl rwshc
O k�twji tÔpoc eÐnai exairetik c spoudaiìthtoc, afoÔ qrhsimopoieÐtai stonupologismì oloklhr¸matoc, rht c sunart sewc.'Estw Iν =
∫ dx(x2+a2)ν , tìte:
Iν =1
a2
[x
(2ν − 2)(x2 + a2)ν−1+
2ν − 3
2ν − 2Iν−1
]
ìtan ν di�foro tou 1.
35
36 KEF�ALAIO 6. ANAGWGIKO�I T�UPOI
6.4 Lumènec Ask seic.
6.1 UpologÐsate to∫
συν5xdx.
LÔsh: Ja qrhsimopoi soume ton sqetikì anagwgikì tÔpo. Jètoume I5 =∫συν5xdx kai èqoume, antikajist¸ntac ston Iν = συνν−1xηµx
ν+ ν−1
νIν−2:
I5 =συν4xηµx
5+
5− 1
5I3
Gia to I3 èqoume:
I3 =συν2xηµx
3+
3− 1
3I1
kaiI1 =
∫συνxdx = ηµx + K
Me antÐstrofh antikat�stash kai ektèlesh twn pr�xewn, èqoume telik�:
I5 =συν4xηµx
5+
4συν2xηµx
15+
8
15ηµx + C
q.e.d.
6.2 UpologÐsate to∫ 1
(x2+3)3dx .
LÔsh: ja qrhsimopoi soume ton anagwgikì tÔpo rht c olokl rwshc.'Estw I3 =
∫ 1(x2+3)3
dx tìte:
I3 =∫ 1
(x2 + (√
3)2)3dx
, opìte α =√
3 kai diadoqik� èqoume:
I3 =1
(√
3)2
[x
(2 · 3− 2)[x2 + (√
3)2]3−1+
2 · 3− 3
2 · 3− 2· I2
]
I2 =1
(√
3)2
[x
(2 · 2− 2)[x2 + (√
3)2]2−1+
2 · 2− 3
2 · 2− 2· I1
]
I1 =∫ 1
x2 + (√
3)2dx =
1√3τoξεϕ
x√3
+ K
Me antÐstrofh antikat�stash kai ektèlesh twn pr�xewn, èqoume telik�:
I3 =x
12(x2 + 3)2+
x
24(x2 + 3)+
1
24√
3τoξεϕ
x√3
+ C
q.e.d.
6.4. LUM�ENES ASK�HSEIS. 37
6.3 UpologÐsate to ∫ 1
(x2 + x + 2)2dx
LÔsh: Ja gr�youme kat' arq�c to x2 + x + 2 san �jroisma tetrag¸nwn:
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 olokl rwma t¸ra gr�fetai:
I2 =∫ 1[(
x + 12
)2+(√
72
)2]2dx
Qrhsimopoi¸ntac ton anagwgikì tÔpo èqoume:
I2 =1(√7
2
)2
x + 12
(2 · 2− 2)[(
x + 12
)2+(√
72
)2]2−1 +
2 · 2− 3
2 · 2− 2· I1
I1 =∫ 1[(
x + 12
)2+(√
72
)2]dx =
2√7τoξεϕ
(2x + 1√
7
)+ K
Me antÐstrofh antikat�stash kai ektèlesh twn pr�xewn, èqoume telik�:
I2 =2x + 1
7(x2 + x + 2)+
4
7√
7τoξεϕ
(2x + 1√
7
)+ C
q.e.d.
6.4 UpologÐsate to
∫ 1
(5x2 + x + 1)3dx
38 KEF�ALAIO 6. ANAGWGIKO�I T�UPOI
LÔsh: To olokl rwma diadoqik� gr�fetai:∫ 1
(5x2 + x + 1)3dx =
∫ 1
53(x2 + 1
5x + 1
5
)3dx =
=1
125
∫ 1(x2 + 1
5x + 1
5
)3dx
Metatrèpoume to x2 + 15x + 1
5se �jroisma tetrag¸nwn :
x2 +1
5x +
1
5= x2 + 2 · 1
2· 1
5x +
(1
10
)2
−(
1
10
)2
+1
5=
=(x +
1
10
)2
+19
100=(x +
1
10
)2
+
(√19
10
)2
To olokl rwma t¸ra gr�fetai:
I3 =1
125
∫ 1[(x + 1
10
)2+(√
1910
)2]3dx
O anagwgikìc tÔpoc ja d¸sei telik�:
I3 =10x + 1
38(5x2 + x + 1)2+
15(10x + 1)
361(5x2 + x + 1)+
300
361√
19τoξεϕ(
10x + 1√19
) + C
q.e.d.
6.5 E�n Iν =∫
ηµνxdx, deÐxate ìti
Iν = −ηµν−1xσυνx
ν+
ν − 1
νIν−2
LÔsh: Ergazìmenoi paragontik� èqoume:
Iν =∫
ηµν−1xηµxdx =∫
ηµν−1x(−συνx)′dx =
= −ηµν−1xσυνx−∫
[−συνx(ηµν−1x)′]dx =
= −ηµν−1xσυνx +∫
[συνx(ν − 1)ηµν−2xσυνx]dx =
= −ηµν−1xσυνx + (ν − 1)∫
[συν2xηµν−2x]dx =
6.4. LUM�ENES ASK�HSEIS. 39
= −ηµν−1xσυνx + (ν − 1)∫
[(1− ηµ2x)ηµν−2x]dx =
= −ηµν−1xσυνx + (ν − 1)∫
[ηµν−2x]dx− (ν − 1)∫
[ηµνx]dx =
= −ηµν−1xσυνx + (ν − 1)Iν−2 − (ν − 1)Iν
LÔnontac aut n thn teleutaÐa exÐswsh wc prìc Iν , paÐrnoume ton zhtoÔmenoanagwgikì tÔpo.
q.e.d.
6.6 E�n Iν =∫(x2 + a2)νdx, deÐxate ìti
Iν =x(x2 + a2)ν
(2ν + 1)+
(2νa2)
(2ν + 1)Iν−1
, ν 6= −12.
LÔsh: Oloklhr¸nontac paragontik�, èqoume diadoqik�:
Iν =∫
[x′(x2 + a2)ν ]dx =
= x(x2 + a2)ν −∫
x[(x2 + a2)ν ]′dx =
= x(x2 + a2)ν −∫
xν(x2 + a2)ν−12xdx
= x(x2 + a2)ν − 2ν∫
(x2 + a2)ν−1x2dx
= x(x2 + a2)ν − 2ν∫
(x2 + a2)ν−1(x2 + a2 − a2)dx
= x(x2 + a2)ν − 2ν∫
(x2 + a2)νdx + 2νa2∫
(x2 + a2)ν−1dx
= x(x2 + a2)ν − 2νIν + 2νa2Iν−1
EpilÔontac thn telik sqèsh, ¸c prìc Iν , paÐrnoume to zhtoÔmeno.
q.e.d.
6.7 BreÐte anagwgikì tÔpo gia to olokl rwma∫(ln x)νdx.
40 KEF�ALAIO 6. ANAGWGIKO�I T�UPOI
LÔsh: 'Estw Iν =∫(ln x)νdx. Diadoqik� èqoume:
Iν =∫
x′(ln x)νdx =
= x(ln x)ν −∫
ν(ln x)ν−1x1
xdx =
= x(ln x)ν − ν∫
(ln x)ν−1dx = x(ln x)ν − νIν−1
o opoÐoc eÐnai kai o zhtoÔmenoc anagwgikìc tÔpoc.
q.e.d.
6.8 BreÐte anagwgikì tÔpo gia to olokl rwma∫ xν
(x2+1)dx.
LÔsh: 'Estw Jν =∫ xν
(x2+1)dx, tìte Jν =
∫ x2
x2+1xν−2dx all� (xν−1)′ = (ν −
1)xν−2 ⇒ xν−2 = (xν−1)′
ν−1antikajist¸ntac, to Jν gÐnetai:
Jν =1
ν − 1
∫ x2
x2 + 1(xν−1)′dx
diadoqik� t¸ra èqoume
Jν =1
ν − 1
∫ [1− 1
x2 + 1
](xν−1)′dx =
=1
ν − 1
∫(xν−1)′dx− 1
ν − 1
∫ (xν−1)′
x2 + 1dx =
=xν−1
ν − 1− 1
ν − 1
∫(ν − 1) · xν−2
x2 + 1dx =
=xν−1
ν − 1−∫ xν−2
x2 + 1dx =
xν−1
ν − 1− Jν−2
o opoÐoc eÐnai kai o zhtoÔmenoc anagwgikìc tÔpoc.
q.e.d.
6.9 BreÐte anagwgikì tÔpo gia to olokl rwma∫ dx
xν√
x2+a, ν 6= 1, a 6=
0.
6.5. ASK�HSEIS PROS EP�ILUSHN 41
LÔsh: 'Estw Jν =∫ dx
xν√
x2+a, diadoqik� èqoume
Jν−2 =∫ dx
xν−2√
x2 + a=∫ x
xν−1√
x2 + adx =
=∫ (
1
x
)ν−1
(√
x2 + a)′dx =
=
√x2 + a
xν−1−∫
(√
x2 + a)(
1
xν−1
)′dx =
=
√x2 + a
xν−1+ (ν − 1)
∫ √x2 + a
xνdx =
=
√x2 + a
xν−1+ (ν − 1)
∫ x2 + a
xν√
x2 + adx =
=
√x2 + a
xν−1+ (ν − 1)
∫ x2
xν√
x2 + adx + (ν − 1)
∫ adx
xν√
x2 + a
kai telik� èqoume
Jν−2 =
√x2 + a
xν−1+ (ν − 1)Jν−2 + a(ν − 1)Jν
kai epilÔontac wc prìc Jν , paÐrnoume ton zhtoÔmeno anagwgikì tÔpo :
Jν = −√
x2 + a
a(ν − 1)xν−1− ν
a(ν − 1)Jn−2
q.e.d.
6.5 Ask seic proc EpÐlushn
6.10 UpologÐsate to olokl rwma∫
ηµ7xdx
Ap: −17ηµ6xσυνx− 6
35ηµ4xσυνx− 24
105ηµ2xσυνx− 48
105+C
6.11 UpologÐsate to olokl rwma∫ 1
(x2+7)2dx
42 KEF�ALAIO 6. ANAGWGIKO�I T�UPOI
Ap: x14(x2+7)
+ 114√
7τoξεϕ( x√
7) + C
6.12 UpologÐsate to olokl rwma∫ 1
(3x2+5)2dx
Ap: x10(3x2+5)
+ 110√
15τoξεϕ(
√35x) + C
6.13 UpologÐsate to olokl rwma∫ 1
(x2−x+7)3dx
Ap: 2x−154(x2−x+7)2
+ 2x−1243(x2−x+7)
+ 4729
√3τoξεϕ
(2x−1√
3
)6.14 UpologÐsate to olokl rwma
∫ 1(7x2−x+2)2
dx
Ap: 14x−155(7x2−x+2)
+ 2855√
55τoξεϕ
(14x−1√
55
)
Kef�laio 7
An�lush se 'Ajroisma Apl¸nKlasm�twn
7.1 Genik�
To phlÐkon dÔo akeraÐwn poluwnÔmwn f(x) kai g(x) kaleÐtai rhtì kl�sma rht sun�rthsh wc proc x. Analutik� gr�foume:
k(x) =f(x)
g(x)=
amxm + am−1xm−1 + · · ·+ a1x + a0
bnxn + bn−1xn−1 + · · ·+ b1x + b0
ìpou am, bn 6= 0 kai ai, bj an koun sto R, i = 1, 2, . . . ,m, j = 1, 2, . . . , n.
7.2 An�lush se Apl� Kl�smata
Se pollèc efarmogèc qrei�zetai na analÔoume mia rht sun�rthsh se �jroi-sma aploÔsterwn klasm�twn. AkoloujoÔme ton parak�tw algìrijmo.
Algìrijmoc an�lushc se �jroisma apl¸n klasm�twn.
B ma 1on. E�n o arijmht c èqei bajmì megalÔtero tou paranomastoÔ, k�-noume thn diaÐresh f(x) : g(x). Qrhsimopoi¸ntac thn tautìthta thc
diaÐreshc èqoume: f(x) = g(x)p(x) + v(x) kai �ra f(x)g(x)
= p(x) + v(x)g(x)
,
ìpou to kl�sma v(x)g(x)
èqei plèon arijmht bajmoÔ mikrìterou tou para-nomastoÔ.
B ma 2on. E�n o arijmht c èqei bajmì mikrìtero tou paranomastoÔ, tìteparagontopoioÔme pl rwc ton paranomast .
43
44 KEF�ALAIO 7. AN�ALUSH SE �AJROISMA APL�WN KLASM�ATWN
B ma 3on. Qrhsimopoi¸ntac ton kat�llhlo tÔpo, exis¸noume to f(x)g(x)
meèna �jroisma apl¸n klasm�twn pou perièqoun stajerèc proc prosdio-rismìn.
B ma 4on. ProsdiorÐzoume tic stajerèc.
7.3 TÔpoi an�lushc se apl� kl�smata
H an�lush sto 3on b ma tou parap�nw algorÐjmou, gÐnetai b�sei twn tÔpwntwn k�twji peript¸sewn (Heavaside).
PerÐptwsh I: E�n to g(x) èqei mìno aplèc pragmatikèc rÐzec p1, p2, . . . , pn,dhlad g(x) = (x− p1)(x− p2) · · · (x− pn) tìte
f(x)
g(x)=
f(x)
(x− p1)(x− p2) · · · (x− pn)=
A1
(x− p1)+
A2
(x− p2)+ · · ·+ An
(x− pn)
ìpou A1, A2, . . . , An stajerèc proc prosdiorismìn.
PerÐptwsh II: E�n to g(x) èqei kai pollaplèc rÐzec, dhlad g(x) = (x −p1)(x− p2)(x− p3)
k · · · (x− pm)l, tìte
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 stajerèc proc prosdiorismìn.
PerÐptwsh III: E�n to g(x) èqei thn morf g(x) = (α1x2 + β1x + γ1) ·
(α2x2 + β2x + γ2)
ϕ tìte
f(x)
g(x)=
f(x)
(α1x2 + β1x + γ1) · (α2x2 + β2x + γ2)ϕ=
=Mx + N
α1x2 + β1x + γ1
+A1x + B1
α2x2 + β2x + γ2
+
+A2x + B2
(α2x2 + β2x + γ2)2+ · · ·+ Aϕx + Bϕ
(α2x2 + β2x + γ2)ϕ
PerÐptwsh IV : 'Otan isqÔoun sugqrìnwc oi peript¸seic I, II, III, tìteefarmìzoume tautìqrona touc antistoÐqouc tÔpouc.
7.4. PROSDIORISM�OS TWN STAJER�WN 45
7.4 Prosdiorismìc twn stajer¸n
Gia na prosdiorÐsoume tic stajerèc qrhsimopoioÔme dÔo mejìdouc: EÐte k�-noume ta kl�smata om¸numa kai exis¸noume touc suntelestèc twn omoiobaj-mÐwn ìrwn twn arijmht¸n, eÐte k�noume ta kl�smata om¸numa kai jètoumeaujaÐretec timèc stic metablhtèc x.
7.5 Lumènec Ask seic
7.1 Na analujeÐ to kl�sma
x2 − x− 1
(x− 1)(x− 2)(x + 3)
se �jroisma apl¸n klasm�twn.
LÔsh: Akolouj¸ntac ton algìrijmo kai thn perÐptwsh I èqoume:
x2 − x− 1
(x− 1)(x− 2)(x + 3)=
A1
x− 1+
A2
x− 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)
exis¸nontac touc arijmhtèc paÐrnoume:
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) gÐnetai: −1 = (−4)A1 ⇒ A1 = 1/4, gia x = 2 h (7.1) gÐnetai:1 = 5A2 ⇒ A2 = 1/5, gia x = −3 h (7.1) gÐnetai: 11 = 20A3 ⇒ A3 = 11/20kai h an�lush telik� eÐnai:
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 analujeÐ to kl�sma
x2 + x + 1
(x + 2)(x + 3)2
se �jroisma apl¸n klasm�twn.
46 KEF�ALAIO 7. AN�ALUSH SE �AJROISMA APL�WN KLASM�ATWN
LÔsh: Apo thn perÐptwsh II èqoume :
x2 + x + 1
(x + 2)(x + 3)2=
A1
x + 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
exis¸nontac touc arijmhtèc èqoume:
x2 + x + 1 = (A1 + B1)x2 + (6A1 + 5B1 + B2)x + (9A1 + 6B1 + 2B2)
sugkrÐnontac touc suntelestèc twn omoiobajmÐwn ìrwn, paÐrnoume:
A1 + B1 = 1
6A1 + 5B1 + B2 = 1
9A1 + 6B1 + 2B2 = 1
kai �raA1 = 3, B1 = −2, B2 = −7
kai h an�lush eÐnai:
x2 + x + 1
(x + 2)(x + 3)2=
3
x + 2− 2
x + 3− 7
(x + 3)2
q.e.d.
7.3 Na analujeÐ se �jroisma apl¸n klasm�twn to kl�sma:
x4 + 1
(x2 − x + 1)3
LÔsh: Epeid èqoume ston paranomast polu¸numo 2ou bajmoÔ, ja qrhsi-mopoi soume ton tÔpo thc perÐptwshc III:
x4 + 1
(x2 − x + 1)3=
A1x + B1
x2 − x + 1+
A2x + B2
(x2 − x + 1)2+
+A3x + B3
(x2 − x + 1)3
k�nontac ta kl�smata om¸numa èqoume, exis¸nontac touc arijmhtèc:
x4 + 1 = (A1x + B1)(x2 − x + 1)2 + (A2x + B2)(x
2 − x + 1) + (A3x + B3)
7.5. LUM�ENES ASK�HSEIS 47
kai k�nontac pr�xeic:
x4+1 = A1x5+(−2A1+B1)x
4+(3A1+A2−2B1)x3+(−2A1−A2+3B1+B2)x
2+
(A1 + A2 + A3 − 2B1 −B2)x + (B1 + B2 + B3)
Exis¸nontac touc suntelestèc twn omoiobajmÐwn ìrwn paÐrnoume to sÔsth-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
Apì ìpou brÐskoume ìti:
A1 = 0, A2 = 2, A3 = −1, B1 = 1, B2 = −1, B3 = 1
Kai h an�lush eÐnai:
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 analujeÐ se apl� kl�smata h rht sun�rthsh:
1
(x2 + 1)(x2 + x)
LÔsh: ParagontopoioÔme pl rwc ton paranomast kai èqoume
1
(x2 + 1)(x2 + x)=
1
(x2 + 1)x(x + 1)
Gia na analÔsoume se apl� kl�smata ja sundu�soume ìlec tic peript¸seic:
1
(x2 + 1)(x2 + x)=
A
x+
B
(x + 1)+
Gx + D
x2 + 1
K�nontac ta kl�smata om¸numa kai exis¸nontac touc arijmhtèc paÐrnoume:
1 = A(x + 1)(x2 + 1) + Bx(x2 + 1) + (Gx + D)x(x + 1) =
48 KEF�ALAIO 7. AN�ALUSH SE �AJROISMA APL�WN KLASM�ATWN
= (A + B + G)x3 + (A + G + D)x2 + (A + B + D)x + A
apì ìpou brÐskoume ìti:
A = 1, B = −1
2, G = −1
2, D = −1
2
kai h zhtoumènh an�lush eÐnai:
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 analujeÐ se apl� kl�smata to kl�sma
x3
x2 − 2x− 3
LÔsh: AfoÔ o arijmht c èqei bajmì megalÔtero apo ton tou paranomastoÔ,k�noume diaÐresh kai èqoume: x3 = (x+2)(x2− 2x− 3)+ (7x+6) to kl�smaja gÐnei:
x3
x2 − 2x− 3=
(x + 2)(x2 − 2x− 3) + 7x + 6
x2 − 2x− 3= (x + 2) +
7x + 6
x2 − 2x− 3
ja analÔsoume t¸ra to kl�sma 7x+6x2−2x−3
,
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)
exis¸nontac touc arijmhtèc èqoume : 7x+6 = A(x+1)+B(x− 3), jètontacx = −1 kai x = 3, brÐskoume A = 27/4, B = 1/4, kai h an�lush telik�gÐnetai :
x3
x2 − 2x− 3= (x + 2) +
27/4
x− 3+
1/4
x + 1
q.e.d.
7.6 UpologÐsate to �jroisma
1
1 · 3+
1
3 · 5+ · · ·+ 1
(2n− 1)(2n + 1)
7.5. LUM�ENES ASK�HSEIS 49
LÔsh: AnalÔoume to kl�sma 1(2n−1)(2n+1)
se �jroisma apl¸n klasm�twn:
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+
−1/2
2n + 1
Diadoqik� t¸ra èqoume:
n = 1 ⇒ 1
1 · 3=
1
2− 1
6
n = 2 ⇒ 1
3 · 5=
1
6− 1
10
n = 3 ⇒ 1
5 · 7=
1
10− 1
14· · ·
n = n ⇒ 1
(2n− 1)(2n + 1)=
1/2
2n− 1+
−1/2
2n + 1
AjroÐzontac kat� mèlh, brÐskoume:
1
1 · 3+
1
3 · 5+ · · ·+ 1
(2n− 1)(2n + 1)=
1
2− 1
2(2n + 1)
q.e.d.
7.7 Na analujeÐ se �jroisma apl¸n klasm�twn to kl�sma
2x2 + 3x− 5
(x− 2)3
LÔsh: Ja lÔsoume thn �skhsh aut me mÐa �llh mèjodo. Diair¸ntac tonarijmht me to (x − 2) èqoume 2x2 + 3x − 5 = (x − 2)(2x + 7) + 9 all�2x + 7 = 2(x − 2) + 11 �ra 2x2 + 3x − 5 = 2(x − 2)2 + 11(x − 2) + 9 kaiepomènwc:
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 KEF�ALAIO 7. AN�ALUSH SE �AJROISMA APL�WN KLASM�ATWN
7.8 Na analujeÐ se �jroisma apl¸n klasm�twn to kl�sma
2x3 − 3x2 + 5x− 3
(x− 1)5
LÔsh: Ja gr�youme ton arijmht wc �jroisma dun�mewn tou (x− 1), qrh-simopoi¸ntac ton tÔpo tou Taylor. GnwrÐzoume ìti to an�ptugma Taylor mekèntro to 1 eÐnai:
f(x) = f(1) + f ′(1)(x− 1)
1!+ f ′′(1)
(x− 1)2
2!+ · · ·
jètontac f(x) = 2x3 − 3x2 + 5x− 3, paÐrnoume:
2x3 − 3x2 + 5x− 3 = 1 + 5(x− 1) + 3(x− 1)2 + 2(x− 1)3
kai to kl�sma gÐnetai:
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)2
q.e.d.
7.9 Na analujeÐ to kl�sma 2x+1x4+1
LÔsh: ParagontopoioÔme 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 kl�sma analÔetai wc ex c:
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 + 1Exis¸nontac touc suntelestèc twn omoiobajmÐwn ìrwn twn arijmht¸n, brÐ-skoume ìti :
A = − 1
2√
2, B =
1
2(1 +
√2), G =
1
2√
2, D =
√2− 2
2√
2
kai h an�lush telik� gÐnetai:
−12√
2x + 1
2(1 +
√2)
(x2 −√
2x + 1)+
12√
2x +
√2−2
2√
2
(x2 +√
2x + 1)
q.e.d.
7.6. ASK�HSEIS PROS EP�ILUSH 51
7.6 Ask seic proc EpÐlush
AnalÔsate se apl� kl�smata tic k�twji rhtèc sunart seic
7.10
2x + 7
(x2 − 4)(x + 1)
Ap: 34(x+2)
+ 112(x−2)
− 53(x−1)
7.11
3x− 1
x2 − 5x + 6
Ap: 8x−3
− 5x−2
7.12
x2 + 11x− 2
(x + 2)(x− 1)(x− 4)
Ap: 19
(29
x−4− 10
x−1− 10
x+2
)7.13
10x2 + 32
x3(x− 4)2
Ap: 1x
+ 1x2 + 2
x3 − 1x−4
+ 3(x−4)2
7.14
x5 + 2)
(x2 + x + 1)3
Ap: (x−2)(x2+x+1)
+ x+3(x2+x+1)2
+ x−1(x2+x+1)3
7.15
x2 − x + 1
(x2 + 1)(x− 1)2
Ap: 12(x2+1)
+ 12(x−1)2
52 KEF�ALAIO 7. AN�ALUSH SE �AJROISMA APL�WN KLASM�ATWN
7.16
x2
(x2 − 2x + 5)2
Ap: 1x2−2x+5
+ 2x−5(x2−2x+5)2
7.17
5x2 − 4
x4 − 5x2 + 4
Ap: 13
(4
x−2− 1
2(x−1)+ 1
2(x+1)− 4
x+2
)7.18
x3
x3 − 3x + 2
Ap: 1 + 13(x−1)2
+ 89(x−1)
− 89(x+2)
7.19
2x3 − 2x− 2
2x2 + x− 6
Ap: −12
+ x + 2x+2
+ 12(2x+3)
7.20
x + 1
x4 − 5x3 + 9x2 − 7x + 2
Ap: 3x−2
− 3x−1
− 3(x−1)2
− 2(x−1)3
7.21
x2 − x + 1
(x− 1)3
Ap: 1x−1
+ 1(x−1)2
+ 1(x−1)3
7.22
8
x8 − 1
7.6. ASK�HSEIS PROS EP�ILUSH 53
Ap: 1x−1
− 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+ x2
9− 80
81(3x+1)2− 4
81(3x+1)
7.25
2x2
3x4 + 2x + 1
Ap:
7.26 Na brejeÐ to �jroisma:
1
2 · 3+
1
3 · 4+ · · ·+ 1
(n + 1)(n + 2)
Ap: 12− 1
n+2
7.27 Na brejeÐ to �jroisma:
1
1 · 3+
1
2 · 4+ · · ·+ 1
n(n + 2)
Ap: 12[ 1n− 1
n+2
7.28 Na brejeÐ to �jroisma:
1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
n(n + 1)(n + 2)
(Upìdeixh: AnalÔsate to kl�sma 1n(n+1)(n+2)
se �jroisma klasm�twn me pa-
ranomastèc n(n + 1) kai (n + 1)(n + 2)).
7.29 DeÐxate ìti:
1
2 · 5+
1
5 · 8+ · · ·+ 1
(3n− 1)(3n + 2)=
1
2
(n
3n + 2
)
Kef�laio 8
Basik� Rht� Oloklhr¸mata
8.1 Genik�
Ta oloklhr¸mata twn k�twji rht¸n sunart sewn qrhsimopoioÔntai gia tonupologismì poluplokotèrwn rht¸n oloklhrwm�twn, all� èqoun kai autìno-mh axÐa.
O1 =∫ dx
(ax + b), O2 =
∫ dx
(ax + b)n, O3 =
∫ dx
ax2 + bx + c
O4 =∫ a1x + b1
a2x2 + b2x + c2
dx, O5 =∫ dx
(a1x2 + b1x + c1)n, O6 =
∫ a1x + b1
(a1x2 + b1x + c1)ndx
Sta trÐa teleutaÐa oloklhr¸mata h diakrÐnousa tou paranomastoÔ jewreÐtaiarnhtik . Ja antimetwpÐsoume to k�je èna xeqwrist�.
8.2 To olokl rwma O1
To olokl rwma autì upologÐzetai b�sei tou tÔpou:
∫ dx
ax + b=
1
aln |ax + b|+ C (8.1)
Prosoq sthn perÐptwsh a < 0.
8.3 To olokl rwma O2
To olokl rwma autì upologÐzetai me antikat�stash, jètontac w = ax + b.
55
56 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA
8.4 To olokl rwma O3
UpologÐzoume thn diakrÐnousa ∆ tou paranomastoÔ kai melet�me tic ex cpeript¸seic:
PerÐptwsh 1h: ∆ > 0.
ParagontopoioÔme ton paranomast ax2 + bx + c = a(x − p1)(x − p2) kaianalÔoume se �jroisma apl¸n klasm�twn:∫ dx
ax2 + bx + c=∫ dx
a(x− p1)(x− p2)=
1
a
∫ A
(x− p1)dx +
1
a
∫ B
(x− p2)dx
PerÐptwsh 2h: ∆ = 0.
ParagontopoioÔme ton paranomast ax2+bx+c = a(x−p)2 kai met� k�noumeantikat�stash, sÔmfwna me thn perÐptwsh O2.
PerÐptwsh 3h: ∆ < 0.Se aut n thn perÐptwsh metatrèpoume ton paranomast se �jroisma tetra-g¸nwn: ax2 + bx + c = a[(x− l)2 + d2] kai met� qrhsimopoioÔme ton tÔpo touτoξεϕ, dhlad èqoume:∫ dx
ax2 + bx + c=∫ dx
a[(x− l)2 + d2]=
=1
a
∫ d(x− l)
(x− l)2 + d2=
1
adτoξεϕ
(x− l
d
)+ C
8.5 To olokl rwma O4
SqhmatÐzoume ston arijmht thn par�gwgo tou paranomastoÔ. Epeita dia-sp�me to kl�sma se duo kl�smata kai epomènwc to arqikì olokl rwma seduo oloklhr¸mata. To pr¸to olokl rwma upologÐzetai me antikat�stash,to deÔtero an kei sthn kathgorÐa O3.
8.6 To olokl rwma O5
Kat' arq�c metatrèpoume thn posìthta a1x2 + b1x + c1 se �jroisma tetra-
g¸nwn: a1x2 + b1x + c1 = a1[(x − s)2 + K2] kai met� qrhsimopoioÔme ton
tÔpo:
In =1
K2
(x− s
(2n− 2)[(x− s)2 + K2]n−1+
2n− 3
2n− 2In−1
), n 6= 1
8.7. TO OLOKL�HRWMA O6 57
ìpou In =∫ 1
[(x−s)2+K2]ndx.
PROSOQH: O arijmìc a1 ja bgeÐ èxw apo to olokl rwma.
8.7 To olokl rwma O6
Me kat�llhlec pr�xeic dhmiourgoÔme ston arijmht thn par�gwgo thc po-sìthtac a1x
2 + b1x + c1. Met� diasp�me to kl�sma se dÔo kl�smata kaiepomènwc to arqikì olokl rwma se dÔo oloklhr¸mata. To pr¸to upologÐ-zetai me antikat�stash, to deÔtero an kei sthn kathgorÐa O5.
8.8 Lumènec Ask seic
8.1 UpologÐsate to∫ dx
2x+6.
LÔsh: B�sei tou tÔpou (8.1), èqoume∫ dx
2x+6= 1
2ln |2x + 6|+ C.
q.e.d.
8.2 UpologÐsate to∫ dx
5−x.
LÔsh: Apì ton tÔpo (8.1) èqoume, gia a = −1,∫ dx
5−xdx = − ln |5− x|+ C.
q.e.d.
8.3 UpologÐsate to∫ dx
(2−x)100.
LÔsh: Ja doulèyoume me antikat�stash. Jètoume w = 2 − x kai �radw = −dx ⇒ dx = −dw. To olokl rwma t¸ra gÐnetai:∫ dx
(2− x)100dx = −
∫ dw
w100dw = −
∫w−100dw =
=−w−100+1
(−100 + 1)+ C = − w−99
(−99)+ C =
1
99· 1
(2− x)99+ C
q.e.d.
8.4 UpologÐsate to olokl rwma∫ dx
(7− 13x)13
58 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA
LÔsh: Jètoume w = 7 − 13x, �ra dw = −13dx ⇒ dx = − 113
dw. Toolokl rwma gÐnetai:∫ dx
(7− 13x)13= −
∫ dw
13w13dw = − 1
13
∫w−13dw =
= − 1
13·(
w−13+1
−13 + 1
)+ C = − 1
13·(
w−12
−12
)+ C =
=1
156· 1
(7− 13x)12+ C
q.e.d.
8.5 ApodeÐxate ton tÔpo:∫ dx
(ax+b)n = − 1a(n−1)(ax+b)n−1 + C.
LÔsh: Me antikat�stash èqoume w = ax + b ⇒ dw = adx ⇒ dx = 1adw kai
to olokl rwma gÐnetai.∫ dx
(ax + b)n=∫ dw
awn=
1
a
∫w−ndw =
=1
a
(w−n+1
−n + 1
)+ C = − 1
a(n− 1)wn−1+ C = − 1
a(n− 1)(ax + b)n−1)+ C
q.e.d.
8.6 UpologÐsate to ∫ dx
x2 − 10x + 21
LÔsh: Blèpoume ìti ∆ = 16 > 0 kai epomènwc paragontopoioÔme ton pa-ranomast : x2 − 10x + 21 = (x − 3)(x − 7). AnalÔoume t¸ra to kl�sma
1x2−10x+21
= 1(x−3)(x−7)
se �jroisma apl¸n klasm�twn. 'Htoi
1
x2 − 10x + 21=−1/4
x− 3+
1/4
x− 7
kai to olokl rwma epomènwc gÐnetai:∫ dx
x2 − 10x + 21=∫ dx
(x− 3)(x− 7)= −1
4
∫ dx
x− 3+
1
4
∫ dx
x− 7=
= −1
4ln |x− 3|+ 1
4ln |x− 7|+ C =
1
4ln∣∣∣∣x− 7
x− 3
∣∣∣∣+ C
q.e.d.
8.8. LUM�ENES ASK�HSEIS 59
8.7 UpologÐsate to olokl rwma∫ dx
4x2 + 4x + 1
LÔsh: ParathroÔme ìti ∆ = 0 kai o paranomast c gr�fetai 4x2 +4x+1 =(2x + 1)2, epomènwc gia to olokl rwma èqoume:∫ dx
4x2 + 4x + 1=∫ dx
(2x + 1)2=
∫ 1
2· dw
w2=
1
2
(w−2+1
−2 + 1
)+ C = −1
2w−1 + C = − 1
2(2x + 1)+ C
q.e.d.
8.8 UpologÐsate to olokl rwma∫ dx
7x2 + 4x + 3
LÔsh: H diakrÐnousa tou paranomastoÔ eÐnai −68 < 0 kai epomènwc ja tonmetatrèyoume se �jroisma tetrag¸nwn. Diadoqik� èqoume:
7x2 + 4x + 3 = 7(x2 + 4
x
7+
3
7
)=
= 7(x2 + 2 · 2
7· x +
4
49− 4
49+
3
7
)= 7
(x +2
7
)2
+
(√17
7
)2
To olokl rwma t¸ra gÐnetai:
∫ dx
7x2 + 4x + 3=∫ 1/7[(
x + 27
)2+(√
177
)2]dx =
1
7
1√
177
τoξεϕ
x + 27√
177
+ C =
√17
17τoξεϕ
(7x + 2√
17
)+ C
q.e.d.
8.9 UpologÐsate to olokl rwma
I =∫ 7x + 9
11x2 + x + 1dx
60 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA
LÔsh: To olokl rwma an kei sthn kathgorÐa O4. SqhmatÐzoume ston a-rijmht thn par�gwgo tou paranomastoÔ (11x2 + x + 1)′ = 22x + 1 kai toolokl rwma mac gÐnetai:
∫ 7x + 9
11x2 + x + 1dx =
1227
∫ 227· (7x + 9)
11x2 + x + 1dx =
=7
22
∫ 22x + 1987
+ 1− 1
11x2 + x + 1dx =
7
22
∫ 22x + 1
11x2 + x + 1dx +
7
22· 191
7
∫ 1
11x2 + x + 1dx (8.2)
To pr¸to olokl rwma upologÐzetai me antikat�stash, en¸ to deÔtero an keisthn kathgorÐa O3. Diadoqik� èqoume
∫ 22x + 1
11x2 + x + 1dx =
∫ 1
wdw = ln |11x2 + x + 1|+ C (8.3)
ìpou w = 11x2 + x + 1, dw = (22x + 1)dx.
∫ 1
11x2 + x + 1dx =
1
11
∫ 1
x2 + 111
x + 111
dx =
=1
11
∫ 1(x + 1
22
)2+(√
4322
)2dx =1
11
1√
4322
τoξεϕ
x + 122√
4322
+ C =
= 2
√43
43τoξεϕ
(22x + 1√
43
)+ C (8.4)
Antikajist¸ntac ta (8.3) kai (8.4) sto (8.2) èqoume telik�:
I =7
22ln |11x2 + x + 1|+ 191
473
√43τoξεϕ
(22x + 1√
43
)+ C
q.e.d.
8.10 UpologÐsate to olokl rwma
I =∫ dx
(11x2 + x + 1)3
8.8. LUM�ENES ASK�HSEIS 61
LÔsh: Gr�foume kat' arq�c to olokl rwma sthn morf
I =∫ dx
113(x2 + x
11+ 1
11
)3 =1
113
∫ dx(x2 + x
11+ 1
11
)3
Gr�foume to x2 + x11
+ 111
se �jroisma tetrag¸nwn x2 + x11
+ 111
=(x + 1
22
)2+(√
4322
)2kai to olokl rwma gÐnetai:
∫ dx(x2 + x
11+ 1
11
)3 =∫ dx((
x + 122
)2+(√
4322
)2)3 =
∫ dX
(X2 + K2)3
ìpou X = x + 1/22 kai K =√
13/22. Ja qrhsimopoi soume t¸ra tonanadromikì tÔpo thc perÐptwshc 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
Kτoξεϕ
X
K
Antikajist¸ntac ìla ta prohgoÔmena antÐstrofa kai lamb�nontac upìyin tictimèc twn X kai K brÐskoume telik� to I.
q.e.d.
8.11 UpologÐsate to olokl rwma
I =∫ x− 7
(x2 − x + 1)2dx
LÔsh: To olokl rwma autì an kei sthn kathgorÐa O6. Ja sqhmatÐsoumeston arijmht thn par�gwgo tou x2 − x + 1, (x2 − x + 1)′ = 2x − 1. To IgÐnetai
I =1
2
∫ 2(x− 7)
(x2 − x + 1)2dx =
1
2
∫ 2x− 1− 13
(x2 − x + 1)2dx =
1
2
∫ 2x− 1
(x2 − x + 1)2dx− 13
2
∫ 1
(x2 − x + 1)2dx (8.5)
62 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA
To pr¸to olokl rwma upologÐzetai me antikat�stash. Jètoume w = x2 −x + 1 ⇒ dw = (2x− 1)dx kai �ra
∫ 2x− 1
(x2 − x + 1)2dx =
∫ 1
w2dw =
∫w−2dw =
w−1
−1+ C = − 1
x2 − x + 1+ C
(8.6)To deÔtero olokl rwma an kei sthn kathgorÐa O5. Gr�foume to x2 − x + 1
se �jroisma tetrag¸nwn: x2 − x + 1 =(x− 1
2
)2+(√
32
)2. To deÔtero
olokl rwma t¸ra gÐnetai:∫ 1
(x2 − x + 1)2dx =
∫ 1(x− 1
2
)2+(√
32
)2dx =∫ 1
(X2 + K2)2dX
ìpou X = x− 12, K =
√3
2. Qrhsimopoi¸ntac t¸ra ton tÔpo thc O5 èqoume:
∫ 1
(X2 + K2)2dX = I2 =
1
K2
X
2(X2 + K2)+
1
2I1
I1 =1
Kτoξεϕ
(X
K
)+ C =
2√3τoξεϕ
(2x− 1√
3
)+ C
∫ 1
(x2 − x + 1)2dx =
4
3
[x− 1/2
2[(x− 1/2)2 + 3/4]
]
+1
2
2√3τoξεϕ
(2x− 1√
3
)+ C (8.7)
Antikajist¸ntac ta apotelèsmata (8.6),(8.7), sto (8.5) kai k�nontac tic pr�-xeic brÐskoume telik� :
I =−1
2(x2 − x + 1)− 13
(x− 1/2)
[(x− 1/2)2 + (3/4)]− 26
3√
3τoξεϕ
(2x− 1)√3
+ C
q.e.d.
8.9 Ask seic Proc EpÐlushn.
8.12 UpologÐsate to olokl rwma∫ dx
3x+7.
Ap: 13ln |3x + 7|+ C.
8.13 UpologÐsate to∫ dx
12−8x.
8.9. ASK�HSEIS PROS EP�ILUSHN. 63
Ap: −18ln |12− 8x|+ C.
8.14 ApodeÐxate ton tÔpo thc perÐptwshc O5.
8.15 UpologÐsate to∫ dx
(2x−9)20.
Ap: − 138· 1
(2x−9)19+ C.
8.16 UpologÐsate to∫ dx
(10−5x)123.
Ap: 15· 1
122· 1
(10−5x)122+ C.
8.17 UpologÐsate to∫ dx
4x2−8x.
Ap: 18ln∣∣∣x−2
x
∣∣∣+ C.
8.18 UpologÐsate to∫ dx
x2−9.
Ap: 16ln∣∣∣x−3x+3
∣∣∣+ C.
8.19 UpologÐsate to∫ dx
a(x2−A).
Ap: 1a√
Aln∣∣∣x−√Ax+
√A
∣∣∣+ C.
8.20 UpologÐsate to∫ dx
2x2−6x−7.
Ap: 12√
23ln∣∣∣2x−3−
√23
2x−3+√
23
∣∣∣+ C.
8.21 UpologÐsate to∫ dx
x2−6x+9.
Ap: − 1x−3
+ C.
8.22 UpologÐsate to∫ dx
x2−4x+4.
Ap: − 1x−2
+ C.
8.23 UpologÐsate to∫ dx
x2+2√
2x+2.
Ap: − 1x+
√2
+ C.
8.24 UpologÐsate to∫ dx
x2−6x+13.
64 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA
Ap: 12τoξεϕ
(x−3
2
)+ C.
8.25 UpologÐsate to∫ dx
2x2+3x+4.
Ap: 2√
2323
τoξεϕ(
4x+3√23
)+ C.
8.26 UpologÐsate to∫ dx
x2−x+1.
Ap: 2√
33
τoξεϕ(
2x−1√3
)+ C.
8.27 UpologÐsate to∫ dx
x2−8x+17.
Ap: τoξεϕ(x− 4) + C.
8.28 UpologÐsate to∫ 7x+8
4x2−16x+52dx.
Ap: 78ln |4x2−16x+52|+11
6τoξεϕ
(x−2
3
)+
C.
8.29 UpologÐsate to∫ 2x−3
4x2+11dx.
Ap: 14ln |4x2 + 11| − 3
2√
11τoξεϕ
(2x√11
)+ C.
8.30 UpologÐsate to∫ dx
(x2+2)3.
Ap: x8(x2+2)2
+ 3x32(x2+2)
+ 332√
2τoξεϕ
(x√2
)+ C.
8.31 UpologÐsate to∫ 3x
(x2+7)2dx.
Ap: − 32(x2+7)
Kef�laio 9
Olokl rwsh Rht¸n Sunart sewn
9.1 Genik�
Jèloume na upologÐsoume oloklhr¸mata thc morf c:
∫ P (x)
Q(x)dx (9.1)
ìpou P (x) kai Q(x) polu¸numo tou x. Autèc oi sunart seic lègontai rhtèckai ta antÐstoiqa oloklhr¸mata rht�.
9.2 H Mèjodoc
Gia ton upologismì tou (9.1) akoloujoÔme thn ex c mèjodo:
BHMA 1on: E�n o bajmìc tou P (x) eÐnai megalÔteroc tou bajmoÔ touQ(x), k�noume thn diaÐresh kai diasp�me to arqikì olokl rwma (9.1),se èna poluwnumikì olokl rwma kai se èna olokl rwma rht c sun�r-thshc me bajmì arijmhtoÔ, mikrìtero tou bajmoÔ tou paranomastoÔ.
BHMA 2on: E�n o bajmìc tou arijmhtoÔ eÐnai mikrìteroc tou bajmoÔ touparanomastoÔ, paragontopoioÔme pl rwc ton paranomast .
BHMA 3on: AnalÔoume to kl�sma P (x)Q(x)
se �jroisma apl¸n klasm�twn.
BHMA 4on: Diasp�me to arqikì olokl rwma (9.1) se epimèrouc basik�rht� oloklhr¸mata, ta opoÐa upologÐzontai kata ta gnwst�.
65
66 KEF�ALAIO 9. OLOKL�HRWSH RHT�WN SUNART�HSEWN
9.3 Lumènec Ask seic
9.1 UpologÐsate to ∫ dx
x3 − 1
LÔsh: ParagontopoioÔme kat' arq�c ton paranomast : x3 − 1 = (x −1)(x2 +x+1) kai analÔoume to kl�sma 1
x3−1se �jroisma apl¸n klasm�twn:
1
x3 − 1=
1
(x− 1)(x2 + x + 1)=
A
x− 1+
Bx + G
x2 + x + 1
Me pr�xeic brÐskoume oti
A =1
3, B = −1
3, G = −2
3
kai to olokl rwma gÐnetai:∫ dx
x3 − 1=
1
3·∫ dx
x− 1− 1
3·∫ x + 2
x2 + x + 1dx
UpologÐzoume ta epimèrouc oloklhr¸mata. To pr¸to eÐnai:∫ dx
x− 1= ln |x− 1|+ C
To deÔtero an kei sthn kathgorÐa O4 kai sqhmatÐzoume ston arijmht thnpar�gwgo tou paranomastoÔ:∫ x + 2
x2 + x + 1dx =
1
2
∫ 2x + 4
x2 + x + 1dx =
1
2
∫ 2x + 1 + 3
x2 + x + 1dx =
=1
2
∫ 2x + 1
x2 + x + 1dx +
3
2
∫ dx
x2 + x + 1=
=1
2
∫ 2x + 1
x2 + x + 1dx +
3
2
∫ dx
[(x + 12)2 + (
√3
2)]2
=
=1
2ln |x2 + x + 1|+ 3
2
(2√3
)τoξεϕ
x + 12√
32
+ C
Antikajist¸ntac èqoume telik�:
∫ dx
x3 − 1=
1
3ln |x− 1| − 1
6ln |x2 + x + 1| −
√3
3τoξεϕ
(2x + 1√
3
)+ C
q.e.d.
9.3. LUM�ENES ASK�HSEIS 67
9.2 UpologÐsate to ∫ dx
x3 − 19x + 30
LÔsh: ParagontopoioÔme kat' arq�c ton paranomast : x3 − 19x + 30 =(x− 2)(x− 3)(x + 5) kai to olokl rwma gÐnetai:∫ dx
x3 − 19x + 30=∫ dx
(x− 2)(x− 3)(x + 5)
AnalÔoume se �jroisma apl¸n klasm�twn:
1
(x− 2)(x− 3)(x + 5)=
A
x− 2+
B
x− 3+
G
x + 5
kai upologÐzoume oti:
A = −1
7, B =
1
8, G =
1
56
kai to olokl rwma metatrèpetai se∫ dx
x3 − 19x + 30= −1
7
∫ dx
x− 2+
1
8
∫ dx
x− 3+
1
56
∫ dx
x + 5=
= −1
7ln |x− 2|+ 1
8ln |x− 3|+ 1
56ln |x + 5|+ C =
=1
56ln
∣∣∣∣∣(x− 3)7(x + 5)
(x− 2)8
∣∣∣∣∣+ C
q.e.d.
9.3 UpologÐsate to olokl rwma:∫ x2 + 6x− 1
x4 + xdx
LÔsh: ParagontopoioÔme pl rwc ton paranomast : x4 + x = x(x3 + 1) =
x(x+1)(x2−x+1). AnalÔoume t¸ra to kl�sma x2+6x−1x4+x
se �jroisma apl¸nklasm�twn:
x2 + 6x− 1
x4 + x=
x2 + 6x− 1
x(x + 1)(x2 − x + 1)=
A
x+
B
x + 1+
Gx + D
x2 − x + 1
Me exÐswsh twn suntelest¸n twn omoiobajmÐwn ìrwn brÐskoume
A = −1, B = 2, G = −1, D = 4
68 KEF�ALAIO 9. OLOKL�HRWSH RHT�WN SUNART�HSEWN
kai to arqikì kl�sma gÐnetai:
x2 + 6x− 1
x4 + x= −1
x+
2
x + 1+
−x + 4
x2 − x + 1
To arqikì olokl rwma t¸ra gÐnetai:
∫ x2 + 6x− 1
x4 + xdx = −
∫ dx
x+ 2
∫ dx
x + 1+∫ −x + 4
x2 − x + 1dx
Ta epimèrouc oloklhr¸mata eÐnai:
∫ dx
x= ln |x|+ C1
∫ dx
x + 1= ln |x + 1|+ C2
∫ −x + 4
x2 − x + 1dx = −
∫ x− 4
x2 − x + 1dx =
= −(
1
2
) ∫ 2x− 8
x2 − x + 1dx = −1
2
∫ 2x− 1− 7
x2 − x + 1dx =
= −1
2
∫ 2x− 1
x2 − x + 1dx +
7
2
∫ dx
(x− 1/2)2 + (√
3/2)=
= −1
2ln |x2 − x + 1|+ 7
2
1√
32
τoξεϕ
x− 1/2√
32
+ C3
Me antikat�stash kai ektèlesh pr�xewn brÐskoume to telikì apotèlesma:
∫ x2 + 6x− 1
x4 + xdx = − ln |x|+ 2 ln |x + 1| − 1
2ln |x2 − x + 1|+
+7
2
1√
32
τoξεϕ
x− 1/2√
32
+ C
q.e.d.
9.4 UpologÐsate to olokl rwma:
∫ x3
x2 + 1dx
9.4. ASK�HSEIS PROS EP�ILUSHN. 69
LÔsh: Epeid o arijmht c èqei bajmì megalÔtero tou paranomastoÔ, jak�noume thn diaÐresh: x3 : x2 + 1 kai brÐskoume phlÐko x kai upìloipo −x,epomènwc x3 = (x2 + 1)x − x. Antikajist¸ntac aut thn èkfrash stonarijmht , to olokl rwma gÐnetai:
∫ x3
x2 + 1dx =
∫ (x2 + 1)x− x
x2 + 1dx =
∫xdx−
∫ x
x2 + 1dx =
=x2
2− 1
2
∫ 2x
x2 + 1dx =
x2
2− 1
2ln |x2 + 1|+ C
q.e.d.
9.5 UpologÐsate to olokl rwma:
∫ x2 + 2x− 1
x2 + 1dx
LÔsh: Epeid o arijmht c èqei ton Ðdio bajmì me ton bajmì tou paranoma-stoÔ, ja k�noume thn diaÐresh. BrÐskoume phlÐko Ðson me 1 kai upìloipo Ðsome 2x− 2, epomènwc x2 + 2x− 1 = (x2 + 1)1 + (2x− 2) kai to olokl rwmagÐnetai:
∫ x2 + 2x− 1
x2 + 1dx =
∫ (x2 + 1) · 1 + (2x− 2)
x2 + 1dx =
=∫
1dx +∫ 2x− 1
x2 + 1dx = x +
∫ 2x
x2 + 1dx− 2
∫ 1
x2 + 1dx =
= x + ln |x2 + 1| − 2τoξεϕx + C
q.e.d.
9.4 Ask seic proc EpÐlushn.
UpologÐsate ta k�twji oloklhr¸mata:
9.6∫ 2−x
5x2+4x−7dx
Ap: 1770
ln |10x− 3| − 3170
ln |10x + 11|+ C
9.7∫ x2−7
x+2dx.
Ap: −3 ln |x + 2|+ x2
2− 2x + C
70 KEF�ALAIO 9. OLOKL�HRWSH RHT�WN SUNART�HSEWN
9.8∫ 2x4
(x2+1)2dx.
Ap: 2x + xx2+1
− 3τoξεϕx + C
9.9∫ dx
x3+1.
Ap: 13ln |x+1|− 1
6ln |x2−x+1|+ 1√
3τoξεϕ
(2x−1√
3
)+C
9.10∫ dx
x3−1.
Ap: 13ln |x−1|− 1
6ln |x2 +x+1|− 1√
3τoξεϕ
(2x+1√
3
)+C
9.11∫ x4−x3−x−1
x2−xdx.
Ap: x3
3+ ln |x| − 2 ln |x− 1|+ C
9.12∫ dx
x4+q4 .
Ap: 12√
2q3
[τoξεϕ
(2x−
√2q√
2q
)+ τoξεϕ
(2x+
√2q√
2q
)]+ 1
4√
2q3 ln∣∣∣ q2+
√2qx+x2
q2−√
2qx+x2
∣∣∣+C
9.13∫ dx
x4+x2+1.
Ap: 12√
3
[τoξεϕ
(2x−1√
3
)+ τoξεϕ
(2x+1√
3
)]+ 1
4ln∣∣∣x2+x+1x2−x+1
∣∣∣+ C
9.14∫ x5dx
x3+x
Ap: x3
x− xτoξεϕx + C
9.15∫ x2+5x−6
x2−7x+9dx.
Ap:
9.16∫ 5x−6
x+7dx
Ap: 5x− 41 ln |x + 7|+ C
9.17∫ x2+2
(x+1)3(x−2)dx.
Ap: 1−2x6(1+x)2
+ 29ln∣∣∣x−2x+7
∣∣∣+ C
9.18∫ x3−7x2+11
(x−2)2(x+1)2dx
Ap: 1x−2
− 13(x+1)
− 109
ln |x− 2|+ 199
ln |x + 1|+ C
9.19∫ 2x
(x2+1)(x2+3)dx
Ap: 12ln∣∣∣x2+1x2+3
∣∣∣+ C
Kef�laio 10
Olokl rwsh Trigwnometrik¸nSunart sewn
10.1 Genik�
Parìti up�rqei genikìc trìpoc gia thn olokl rwsh twn trigwnometrik¸nsunart sewn, ton opoÐo kai ja exet�soume sto tèloc tou kefalaÐou, merikèctrigwnometrikèc sunart seic oloklhr¸nontai eukolìtera. Ja xekin soumeto kef�laio me thn parousi�sh aut¸n twn eidik¸n morf¸n.
10.2 Trigwnometrik� Ginìmena
Metatrèpoume ta ginìmena se ajroÐsmata, b�sei twn tÔpwn:
∫ηµ(Ax)συν(Bx)dx =
1
2
∫ηµ(Ax + Bx)dx +
1
2
∫ηµ(Ax−Bx)dx
∫συν(Ax)ηµ(Bx)dx =
1
2
∫ηµ(Ax + Bx)dx− 1
2
∫ηµ(Ax−Bx)dx
∫συν(Ax)συν(Bx)dx =
1
2
∫συν(Ax + Bx)dx +
1
2
∫συν(Ax−Bx)dx
∫ηµ(Ax)ηµ(Bx)dx =
1
2
∫συν(Ax−Bx)dx− 1
2
∫συν(Ax + Bx)dx
71
72KEF�ALAIO 10. OLOKL�HRWSH TRIGWNOMETRIK�WN SUNART�HSEWN
10.3 Olokl rwsh peritt¸n dun�mewn, trigwno-metrik¸n sunart sewn
IsqÔoun oi k�twji empeirikoÐ kanìnec:
• To olokl rwma∫
ηµ2n+1xdx upologÐzetai me thn antikat�stash w =
συνx.
• To olokl rwma∫
συν2n+1xdx upologÐzetai me thn antikat�stash
w = ηµx.
• To olokl rwma∫
εϕ2n+1xdx upologÐzetai me thn metatrop εϕx =
ηµx
συνxkai thn antikat�stash w = συνx.
• To olokl rwma∫
σϕ2n+1xdx upologÐzetai me thn metatrop σϕx =
συνxηµx
kai thn antikat�stash w = ηµx.
10.4 Olokl rwsh artÐwn dun�mewn, trigwno-metrik¸n sunart sewn
IsqÔoun oi k�twji empeirikoÐ kanìnec :
• Gia na upologÐsoume to olokl rwma∫
ηµ2νxdx qrhsimopoioÔme to-
n tÔpo: ηµ2x = 1−συν2x2
, gia na p�roume oloklhr¸mata mikrotèroubajmoÔ.
• Gia na upologÐsoume to olokl rwma∫
συν2νxdx qrhsimopoioÔme ton
tÔpo: συν2x = 1+συν2x2
gia na p�roume oloklhr¸mata mikrotèroubajmoÔ.
• Gia na p�roume to olokl rwma∫
εϕ2νxdx metatrèpoume to εϕ2νx se
ginìmeno εϕ2ν−2x·εϕ2x kai qrhsimopoioÔme ton tÔpo εϕ2x = 1συν2x
−1.ProkÔptoun dÔo oloklhr¸mata, to pr¸to upologÐzetai me thn antikat�-stash w = εϕx kai to deÔtero eÐnai ìmoio me to arqikì all� mikrotèroubajmoÔ.
10.5. OLOKLHR�WMATA GINOM�ENWNDUN�AMEWN TRIGWNOMETRIK�WN SUNART�HSEWN73
• Gia na p�roume to olokl rwma∫
σϕ2νxdx paragontopoioÔme σϕ2νx
sÔmfwna me thn sqèsh σϕ2ν−2x · σϕ2x kai qrhsimopoioÔme ton tÔpoσϕ2x = 1
ηµ2x− 1. ProkÔptoun dÔo oloklhr¸mata, to pr¸to upologÐ-
zetai me thn antikat�stash w = σϕx kai to deÔtero eÐnai ìmoio me toarqikì all� mikrotèrou bajmoÔ.
10.5 Oloklhr¸mata ginomènwn dun�mewn tri-gwnometrik¸n sunart sewn
IsqÔoun oi k�twji empeirikoÐ kanìnec:
• Gia na upologÐsoume oloklhr¸mata thc morf c∫
ηµθxσυν2k+1xdx ,
θ, k akèraioi, ekfr�zoume to olokl rwma sunart sei tou ηµx kai qrh-simopoioÔme thn antikat�stash w = ηµx
• Gia na upologÐsoume oloklhr¸mata thc morf c∫
ηµθxσυν2kxdx , θ, k
akèraioi, ekfr�zoume to olokl rwma sunart sei tou συνx kai qrhsi-mopoioÔme thn antikat�stash w = συνx.
10.6 Oloklhr¸mata phlÐkwn dun�mewn trigw-nometrik¸n sunart sewn
• Gia na upologÐsoume oloklhr¸mata thc morf c∫ ηµθx
συνρxdx ρ, θ -
jetikoÐ akèraioi, ta metatrèpoume, qrhsimopoi¸ntac ton tÔpo ηµ2x +συν2x = 1, se olokl rwma enìc mìno trigwnometrikoÔ arijmoÔ kaidiasp�me to kl�sma.
• Gia na upologÐsoume oloklhr¸mata thc morf c∫ 1
ηµρxσυνθxdx ρ, θ
- jetikoÐ akèraioi, antikajistoÔme thn mon�da tou arijmhtoÔ me thnposìthta ηµ2x + συν2x = 1 kai diasp�me to kl�sma.
74KEF�ALAIO 10. OLOKL�HRWSH TRIGWNOMETRIK�WN SUNART�HSEWN
10.7 Genikìc trìpoc olokl rwshc trigwnome-trik¸n sunart sewn
'Ena opoiod pote trigwnometrikì olokl rwma thc morf c∫
f(ηµx, συνx, εϕx)dx
metatrèpetai se rhtì, me thn bo jeia twn metasqhmatism¸n:
dx =2dt
t2 + 1, ηµx =
2t
1 + t2, συνx =
1− t2
1 + t2, εϕx =
2t
1− t2
Oi metasqhmatismoÐ autoÐ prokÔptoun apo thn sqèsh t = εϕx2. H mèjodoc
aut , parìti antimetwpÐzei opoiod pote trigwnometrikì olokl rwma, odhgeÐsun jwc se perÐploka rht� oloklhr¸mata kai gia autì kalì eÐnai na apo-feÔgetai.
10.8 Lumènec Ask seic
10.1 UpologÐsate to olokl rwma∫
συν(5x)συν(2x)dx.
LÔsh: Apì touc gnwstoÔc trigwnometrikoÔc tÔpouc èqoume:
συν(5x)συν(2x) =1
2συν(5x + 2x) +
1
2συν(5x− 2x) =
=1
2συν(7x) +
1
2συν(3x)
kai �ra ∫συν(5x)συν(2x)dx =
1
2
[∫συν(7x)dx +
∫συν(3x)dx
]=
=1
14ηµ(7x) +
1
6ηµ(3x) + C
q.e.d.
10.2 UpologÐsate to olokl rwma∫
συν3xdx.
LÔsh: Jètoume w = ηµx kai �ra dwdx
= συνx ⇒ dx = dwσυνx
. Diadoqik�t¸ra èqoume: ∫
συν3xdx =∫
συν3x1
συνxdw =
=∫
συν2xdw =∫
(1− ηµ2x)dw =∫
(1− w2)dw =
10.8. LUM�ENES ASK�HSEIS 75
∫(1)dw −
∫w2dw = w − w3
3+ C = ηµx− ηµ3x
3+ C
q.e.d.
10.3 UpologÐsate to olokl rwma∫
εϕ5xdx.
LÔsh: Ja qrhsimopoi soume thn antikat�stash w = συνx. Diadoqik�èqoume: ∫
εϕ5xdx =∫ ηµ5x
συν5xdx =
∫ ηµ5x
w5·(−1
ηµx
)dw =
= −∫ ηµ4x
w5dw = −
∫ (1− συν2x)2
w5dw = −
∫ (1− w2)2
w5dw =
= −∫ dw
w5− 2
∫ dw
w3+∫ dw
wdw =
1
4w4− 1
w2+ ln w + C =
=1
4συν4x− 1
συν2x+ ln |συνx|+ C
q.e.d.
10.4 UpologÐsate to olokl rwma∫
συν4xdx.
LÔsh: Diadoqik� èqoume:
∫συν4xdx =
∫(συν2x)2dx =
∫ (1 + συν(2x)
2
)2
dx =
=1
4
∫[1 + συν2(2x) + 2συν(2x)]dx =
1
4
∫1dx +
2
4
∫συν(2x)dx+
+1
4
∫συν2(2x)dx =
x
4+
1
4· 1
2· ηµ(2x) +
1
4
∫ 1 + συν(4x)
2dx =
=x
4+
1
4ηµ(4x) +
1
8
∫1dx +
1
8
∫συν(4x)dx =
=x
4+
1
4ηµ(4x) +
x
8+
1
32ηµ(4x) + C
q.e.d.
10.5 UpologÐsate to olokl rwma∫
σϕ4xdx.
76KEF�ALAIO 10. OLOKL�HRWSH TRIGWNOMETRIK�WN SUNART�HSEWN
LÔsh: Diadoqik� èqoume:
∫σϕ4xdx =
∫σϕ2x · σϕ2xdx =
∫σϕ2x
(1
ηµ2x− 1
)dx =
=∫ σϕ2x
ηµ2xdx−
∫σϕ2xdx
To pr¸to olokl rwma upologÐzetai me antikat�stash w = σϕx ⇒ dw =− dx
ηµ2xkai �ra:
∫ σϕ2x
ηµ2xdx =
∫(−w2)dw = −w3
3+ C1 = −σϕ3x
3+ C1
Gia to deÔtero olokl rwma èqoume:
∫σϕ2xdx =
∫ (1
ηµ2x− 1
)dx =
∫ dx
ηµ2x−∫
1dx = −σϕx− x + C2
kai telik� ∫σϕ4xdx = −σϕ3x
3− σϕx− x + C
q.e.d.
10.6 UpologÐsate to olokl rwma∫
ηµ3xσυν2xdx.
LÔsh: QrhsimopoioÔme thn antikat�stash w = συνx kai èqoume dx = − dwηµx
kai �ra ∫ηµ3x · συν2xdx =
∫ηµ2x · ηµx · συν2xdx =
=∫
(1− συν2x) · ηµx · συν2x ·(− 1
ηµx
)dw =
−∫
(1− w2)w2dw =∫
(w4 − w2)dw =w5
5− w3
3+ C =
=1
5συν5x− 1
3συν3x + C
q.e.d.
10.7 UpologÐsate to olokl rwma
∫ ηµ5x
συν2xdx
10.8. LUM�ENES ASK�HSEIS 77
LÔsh: QrhsimopoioÔme thn antikat�stash w = συνx ⇒ dx = − dwηµx
kai�ra ∫ ηµ5x
συν2xdx =
∫ ηµ5x
w2·(− 1
ηµx
)dw =
= −∫ ηµ4x
w2dw = −
∫ (1− συν2x)2
w2dw =
= −∫ (1− w2)2
w2dw = −
∫ (1− 2w2 + w4
w2
)dw =
= −∫ 1
w2dw + 2
∫(1)dw −
∫w2dw =
=1
w+ 2w − w3
3+ C =
1
συνx+ 2συνx− συν3x
3+ C
q.e.d.
10.8 UpologÐsate to olokl rwma∫ dx
ηµ2xσυν2x
LÔsh: SqhmatÐzoume ston arijmht thn posìthta ηµ2x + συν2x = 1 kaièqoume: ∫ 1
ηµ2xσυν2xdx =
∫ ηµ2x + συν2x
ηµ2xσυν2xdx =
=∫ ηµ2x
ηµ2xσυν2xdx +
∫ συν2x
ηµ2xσυν2xdx =
=∫ dx
συν2x+∫ dx
ηµ2x= εϕx− σϕx + C
q.e.d.
10.9 UpologÐsate to olokl rwma∫ dx
1 + συνx
LÔsh: Ja qrhsimopoi soume thn mèjodo thc genik c trigwnometrik c anti-kat�stashc. Jètoume συνx = 1−t2
1+t2, dx = 2dt
1+t2, ìpou t = εϕ(x
2) kai èqoume:
∫ dx
2 + συνx=∫ 2
1+t2
5 + 1−t2
1+t2
dt =
78KEF�ALAIO 10. OLOKL�HRWSH TRIGWNOMETRIK�WN SUNART�HSEWN
=∫ 2
6 + 4t2dt =
∫ dt
2t2 + 3
to teleutaÐo olokl rwma ja gÐnei:∫ dt
2t2 + 3dt =
1
2
∫ dt
t2 + 32
dt =
=1
2
∫ dt
t2 + (√
32)2
=1
2· 1√
32
τoξεϕ
t√32
+ C =
=
√2
2√
3τoξεϕ
(t√
2√3
)+ C
kai me antÐstrofh antikat�stash èqoume telik�:
∫ dx
1 + συνx=
√2
2√
3τoξεϕ
εϕ(
x2
)√2
√3
+ C
q.e.d.
10.10 UpologÐsate to olokl rwma∫ dx
2 + 3ηµx + συνx
LÔsh: Me antikat�stash èqoume:
∫ 21+t2
2 + 3 2tt2+1
+ 1−t2
1+t2
dt =
= 2∫ dt
2t2 + 2 + 6t + 1− t2=∫ dt
t2 + 6t + 3
AnalÔoume to kl�sma 1t2+6t+3
se �jroisma apl¸n klasm�twn:
1
t2 + 6t + 3= − 1
2√
6(−3 +√
6− t)− 1
2√
6(t + 3 +√
6)
kai to olokl rwma gÐnetai:
2∫ dt
t2 + 6t + 3= − 1√
6
∫ dt
−3 +√
6− t− 1√
6
∫ dt
t + 3 +√
6=
1√6
ln | − 3 +√
6− t| − 1√6
ln |t + 3 +√
6|+ C =
10.9. ASK�HSEIS PROS EP�ILUSHN 79
=1√6
ln
∣∣∣∣∣−3 +√
6− t
t + 3 +√
6
∣∣∣∣∣+ C
kai me antÐstrofh antikat�stash èqoume telik�:
∫ dx
2 + 3ηµx + συνx=
1√6
ln
∣∣∣∣∣−3 +√
6− εϕ(x2)
εϕ(x2) + 3 +
√6
∣∣∣∣∣+ C
q.e.d.
10.9 Ask seic proc EpÐlushn
UpologÐsate ta k�twji oloklhr¸mata:
10.11∫
ηµ(x2)συν(x
2)dx
Ap: −35συν(5
6x)− 3συν(x
6) + C
10.12∫
συνxηµ(nx)dx
Ap: − 12(n+1)
συν((n + 1)x) + 12(n+1)
συν((1− n)x) + C
10.13∫
ηµ(10x)ηµ(6x)dx
Ap: 18ηµ(4x) + 1
32ηµ(16x) + C
10.14∫
συνxσυν(nx)dx
Ap: 12(n+1)
ηµ((n + 1)x) + 12(1−n)
ηµ((1− n)x) + C
10.15∫
εϕ3xdx
Ap: ln |συνx|+ 12συν2x
+ C
10.16∫
εϕ4xdx
Ap: 13εϕ3x− εϕx + x + C
10.17∫
σϕ6xdx
Ap: −15σϕ5x + 1
3σϕ3x− σϕx− x + C
10.18∫ dx
ηµ2συν3x
80KEF�ALAIO 10. OLOKL�HRWSH TRIGWNOMETRIK�WN SUNART�HSEWN
Ap: εϕx
2συνx+ 1
2ln∣∣∣ 1συνx
+ εϕx∣∣∣+ ∫ dx
συν3x
10.19∫ dx
5−4συνx
Ap: 23τoξεϕ(3εϕ(x
2)) + C
10.20∫ dx
ηµx
Ap: ln∣∣∣ 1ηµx
− σϕx∣∣∣+ C
10.21∫ dx
συνx
Ap: ln∣∣∣ 1συνx
+ εϕx∣∣∣+ C
10.22∫ dx
ηµ3x
Ap: − σϕx
2ηµx+ 1
2ln∣∣∣ 1ηµx
− σϕx∣∣∣+ C
10.23∫ dx
συν3x
Ap: εϕx
2συνx+ 1
2ln∣∣∣ 1συνx
+ εϕx∣∣∣+ C
10.24∫ σϕ3x
ηµ4(3x)dx
Ap: −16σϕ2(3x)− 1
12σϕ4(3x) + C
10.25∫ dx
1−ηµ(x2)
Ap: 2εϕ(
x2
+ 1ηµ(x
2)
)+ C
10.26∫ dx
1+ηµx−συνx
Ap: ln∣∣∣∣ εϕ(x
2)
1+εϕ(x2)
∣∣∣∣+ C
10.27∫ dx
1+συν(3x)
Ap: 1−συν(3x)3ηµ(3x)
+ C
10.28∫ dx
1−2ηµx
10.9. ASK�HSEIS PROS EP�ILUSHN 81
Ap:√
33
ln∣∣∣∣εϕ(x
2)−2−
√3
εϕ(x2)−2+
√3
∣∣∣∣+ C
10.29∫ ηµx
1+ηµ2xdx
Ap:√
24
ln∣∣∣∣εϕ2(x
2)+3−2
√2)
εϕ2(x2)+3+2
√2
∣∣∣∣+ C
10.30∫ dx
2+ηµxdx
Ap: 2√3τoξεϕ
(2εϕ(x
2+1)√
3
)+ C
Kef�laio 11
Olokl rwsh Arr twnSunart sewn I
Sto kef�laio autì ja asqolhjoÔme me thn olokl rwsh arr twn sunart se-wn me prwtob�jmio upìrrizo.
11.1 Oloklhr¸mata thc morf c:
∫f
(x, n
√αx + β
γx + δ
)dx ,
α
γ6= β
δ
Ta oloklhr¸mata aut� upologÐzontaai me thn antikat�stash ω = n
√αx+βγx+δ
,metatrepìmena se rht�.
11.2 Oloklhr¸mata thc morf c:
∫f
(x, n1
√αx + β
γx + δ, n2
√αx + β
γx + δ
)dx ,
α
γ6= β
δ
Gia na upologÐsoume aut� ta oloklhr¸mata, metatrèpoume tic eterob�jmiecrÐzec omoiob�jmiec, me thn bo jeia tou elaqÐstou koinoÔ pollaplasÐou n twn
n1, n2 kai met� qrhsimopoioÔme thn antikat�stash ω = n
√αx+βγx+δ
. H antikat�-stash aut metasqhmatÐzei to arqikì olokl rwma se rhtì.
83
84 KEF�ALAIO 11. OLOKL�HRWSH ARR�HTWN SUNART�HSEWN I
11.3 Oloklhr¸mata thc morf c:
∫f(x, n1
√xm1 , n2
√xm2 , . . . , nk
√xmk
)dx
Gia na upologÐsoume aut� ta oloklhr¸mata metatrèpoume tic eter¸numecrÐzec se om¸numec, me thn bo jeia tou elaqÐstou koinoÔ pollaplasÐou n twnt�xewn twn riz¸n n1, n2, . . . , nk kai met� qrhsimopoioÔme ton metasqhmatismìn√
x = ω, gia na metatrèyoume to olokl rwma se rhtì.
11.4 Lumènec Ask seic
11.1 UpologÐsate to olokl rwma:∫ dx
(x + 2)√
x− 7
LÔsh: QrhsimopoioÔme thn antikat�stash:√
x− 7 = ω kai èqoume diado-qik�: x = ω2 + 7 ⇒ dx = 2ωdω, kai to olokl rwma gÐnetai:∫ dx
(x + 2)√
x− 7=∫ 2ωdω
(ω2 + 7 + 2)ω=
= 2∫ dω
ω2 + 9= 2
∫ dω
ω2 + 32=
= 2 · 1
3τoξεϕ
(ω
3
)+ C =
2
3τoξεϕ
(√x− 7
3
)+ C
q.e.d.
11.2 UpologÐsate to olokl rwma:∫ 2x− 33√
x− 2dx
LÔsh: Jètoume 3√
x− 2 = ω kai èqoume: x = ω3 + 2 kai dx = 3ω2dω. Toolokl rwma t¸ra diadoqik� gÐnetai:∫ 2x− 3
3√
x− 2dx =
∫ 2(ω3 + 2)− 3
ω· 3ω2dω =
∫3ω(2ω3 + 1)dω =
=∫
(6ω4 + 3ω)dω =6
5ω5 +
3
2ω2 + C =
6
53
√(x− 2)5 +
3
23
√(x− 2)2 + C
q.e.d.
11.4. LUM�ENES ASK�HSEIS 85
11.3 UpologÐsate to olokl rwma:
∫ 1
x(x− 1)3
√x
x− 1dx
LÔsh: Ja qrhsimopoi soume thn antikat�stash: 3
√x
x−1= ω ap' ìpou èqoume
x = ω3
ω3−1kai me parag¸gish paÐrnoume dx
dω= −3ω2
(ω3−1)2⇒ dx = −3ω2
(ω3−1)2dω. To
olokl rwma t¸ra diadoqik� gÐnetai:
∫ 1
x(x− 1)3
√x
x− 1dx =
∫ 1ω3
ω3−1
(ω3
ω3−1− 1
) · ω · −3ω2
(ω3 − 1)2dω =
=∫ −3ω3
ω3dω = −3
∫dω = −3ω + C = −3 3
√x
x− 1+ C
q.e.d.
11.4 UpologÐsate to olokl rwma:
I =∫ 1− 3
√x + 1√
x + 1 + 3√
x + 1dx
LÔsh: Metatrèpoume ta eterob�jmia rizik� se omoiob�jmia kai to olokl -rwma gÐnetai:
I =∫ 1− 6
√(x + 1)2
6
√(x + 1)6 + 6
√(x + 1)2
dx
QrhsimopoioÔme t¸ra thn antikat�stash ω = 6√
x + 1, apì ìpou x = ω6 − 1kai dx = 6ω5dω. 'Eqoume t¸ra diadoqik�:
I =∫ 1− ω2
ω3 + ω2· 6ω5dω =
= −∫ (ω − 1)(ω + 1)
ω2(ω + 1)· 6ω5dω = −6
∫(ω − 1)ω2dω =
= −6∫
(ω4 − ω3)dω = −6
5ω5 +
6
4ω4 + C =
= −6
5( 6√
x + 1)5 +3
2( 6√
x + 1)4 + C
q.e.d.
86 KEF�ALAIO 11. OLOKL�HRWSH ARR�HTWN SUNART�HSEWN I
11.5 UpologÐsate to olokl rwma:
I =∫ √
x4√
x3 + 1dx
LÔsh: Metatrèpoume ta eterob�jmia rizik� se omoiob�jmia kai èqoume:
I =∫ 4
√x2
4√
x3 + 1dx
Jètoume t¸ra ω = 4√
x,⇒ dx = 4ω3dω kai to olokl rwma diadoqik� gÐnetai:
I =∫ ω2
ω3 + 1· 4ω3dω = 4
∫ ω5
ω3 + 1dω =
= 4∫ (
ω − fracω2ω3 + 1)dω = 4
∫ωdω − 4
∫fracω2ω3 + 1dω =
=4
34√
x3 − 4
3ln | 4√
x3 + 1|+ C
q.e.d.
11.5 Ask seic proc EpÐlush
UpologÐsate ta k�twji oloklhr¸mata:
11.6∫ dx√
3x−5
Ap: 23
√3x− 5 + C.
11.7∫ 1−
√3x+2
1+√
3x+2dx
Ap: −13(3x + 2) + 4
3
√3x + 2 − 4
3ln |√
3x + 2 +1|+ C.
11.8∫ 1
x
√1−x1+x
dx
Ap: 2τoξεϕ(√
1−x1+x
)+ ln
∣∣∣√1−x−√
1+x√1−x+
√1+x
∣∣∣+ C.
11.9∫ √
x−23√
x2
4√xdx
Ap: 45x15/12 − 24
17x17/12 + C.
11.5. ASK�HSEIS PROS EP�ILUSH 87
11.10∫ dx
x√
1−x
Ap: ln∣∣∣1−√1−x1+
√1−x
∣∣∣+ C.
11.11∫ dx
3+√
x+2
Ap: 2√
x + 2− 6 ln(3 +√
x + 2) + C.
11.12∫ √3x+2
x−3
Ap:
11.13∫ 1
x2
√1−x1+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+ 4√x−2
3√x−2dx
Ap: