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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.


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.


2.1 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


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


11x2e2x+7dx.

Ap: e2x+7(114− 11

2x + 11

2x2) + C


6x3αxdx.

Ap: 6αx

ln a(−3

2+ 2x− 3

2x2 + x3) + C


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.


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.


(3−2x)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.


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


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.


x(ln x+3).


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.


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


3.17∫ 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


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


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 UpologÐsate to olokl rwma:∫ dx

x2+4.

LÔsh: Diadoqik� èqoume:∫ dx

x2 + 4=∫ dx

x2 + 22=

1

2τoξεϕ

x

2+ C

q.e.d.


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


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.


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


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.


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.

22 KEF�ALAIO 4. OLOKLHR�WMATA M�ESW THS τOξεϕX

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′.


5.1 UpologÐsate to olokl rwma:∫

xσυνxdx.

LÔsh: Diadoqik� èqoume:∫x(ηµx)′dx = xηµx−

∫(x)′ηµxdx =

= xηµx−∫

ηµxdx = xηµx + συνx + C

q.e.d.


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.


x2ηµxdx.

LÔsh: Diadoqik� èqoume:∫x2ηµxdx =

∫x2(−συνx)′dx = −x2συνx +

∫(x2)′συνxdx =


= −x2συνx + 2∫

xσυνxdx = −x2συνx + 2∫

x(ηµx)′dx =

= −x2συνx + 2xηµx− 2∫

(x)′ηµxdx =

= −x2συνx + 2xηµx− 2∫

ηµxdx = −x2συνx + 2xηµx + 2συνx + C

q.e.d.


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.


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.



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.


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


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.


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


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.


5.11 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.


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


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


5.27∫

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


6.1 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


I3 =x

12(x2 + 3)2+

x

24(x2 + 3)+

1

24√

3τoξεϕ

x√3

+ C

q.e.d.


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


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


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 =


= −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.


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.


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.



ηµ7xdx

Ap: −17ηµ6xσυνx− 6

35ηµ4xσυνx− 24

105ηµ2xσυνx− 48

105+C

6.11 UpologÐsate to olokl rwma∫ 1

(x2+7)2dx


Ap: x14(x2+7)

+ 114√

7τoξεϕ( x√

7) + C


(3x2+5)2dx

Ap: x10(3x2+5)

+ 110√

15τoξεϕ(

√35x) + C


(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.


7.1 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



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)


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) =


= (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)


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.


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


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


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


1

1 · 3+

1

2 · 4+ · · ·+ 1

n(n + 2)

Ap: 12[ 1n− 1

n+2


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.


To olokl rwma autì upologÐzetai me antikat�stash, jètontac w = ax + b.

55

56 KEF�ALAIO 8. BASIK�A RHT�A OLOKLHR�WMATA


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


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.


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.


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.


8.1 UpologÐsate to∫ dx

2x+6.

LÔsh: B�sei tou tÔpou (8.1), èqoume∫ dx

2x+6= 1

2ln |2x + 6|+ C.

q.e.d.


5−x.

LÔsh: Apì ton tÔpo (8.1) èqoume, gia a = −1,∫ dx

5−xdx = − ln |5− x|+ C.

q.e.d.


(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


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.



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.


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


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.


I =∫ dx

(11x2 + x + 1)3


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.


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)


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.


3x+7.

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


12−8x.

8.9. ASK�HSEIS PROS EP�ILUSHN. 63

Ap: −18ln |12− 8x|+ C.

8.14 ApodeÐxate ton tÔpo thc perÐptwshc O5.


(2x−9)20.

Ap: − 138· 1

(2x−9)19+ C.


(10−5x)123.

Ap: 15· 1

122· 1

(10−5x)122+ C.


4x2−8x.

Ap: 18ln∣∣∣x−2

x

∣∣∣+ C.


x2−9.

Ap: 16ln∣∣∣x−3x+3

∣∣∣+ C.


a(x2−A).

Ap: 1a√

Aln∣∣∣x−√Ax+

√A

∣∣∣+ C.


2x2−6x−7.

Ap: 12√

23ln∣∣∣2x−3−

√23

2x−3+√

23

∣∣∣+ C.


x2−6x+9.

Ap: − 1x−3

+ C.


x2−4x+4.

Ap: − 1x−2

+ C.


x2+2√

2x+2.

Ap: − 1x+

√2

+ C.


x2−6x+13.


Ap: 12τoξεϕ

(x−3

2

)+ C.


2x2+3x+4.

Ap: 2√

2323

τoξεϕ(

4x+3√23

)+ C.


x2−x+1.

Ap: 2√

33

τoξεϕ(

2x−1√3

)+ C.


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.


(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)


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



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.



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


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.


∫ 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.


∫ 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


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.


συν2n+1xdx upologÐzetai me thn antikat�stash

w = ηµx.


εϕ2n+1xdx upologÐzetai me thn metatrop εϕx =

ηµx

συνxkai thn antikat�stash w = συνx.


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.


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.



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.


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 =


∫(1)dw −

∫w2dw = w − w3

3+ C = ηµx− ηµ3x

3+ C

q.e.d.


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.


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.


σϕ4xdx.


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.


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.


∫ ηµ5x

συν2xdx


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.


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.


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 =


=∫ 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.


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 =


=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.



10.11∫

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


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


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�.


∫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


∫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.



(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.



∫ 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.


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


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


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.


11.10∫ 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:

kef laio 1 to aìristo olokl rwma - eap.gredu.eap.gr/pli/pli12/shmeiwseis/oloklhrwmata_2.pdf2 kef...

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