integracija nekih trig. funkcija

7/25/2019 Integracija Nekih Trig. Funkcija

1/8

1 matematiki praktikum

INTEGRACIJA NEKIH TIPOVA

TRIGONOMETRIJSKIH FUNKCIJA

Integrali oblika

dxxxf

cos,sin rjeavaju se smjenom

2

2 2

2

2

2 1sin ,cos

1 1

2,

2 1

xtg t

t tx x

t t

x dtarctgt dx

t

Dokaz:

2 2 22

22 2 2

cos sin 1 12 2 2cos1cos sin 1

2 2 2

x x xtg

txx x x t

tg

,2

2 2 2

2sin cos 2 22 2 2sin1sin cos 1

2 2 2

x x xtg

txx x x t

tg

2, , 2

2 2 1

x x dttg t arctgt dx

t

ZADACI:

1.

Zadatak:Izraunati integrale:

a)sin

dxI

x b) cos

dxI

x c) 1 sin

dxI

x

d)1

5 sin 3cosI dx

x x

Rjeenje:

a)2

2

2

1 ln ln2 2

1

dt

dt xtI t C tg Ct t

t

b) ln ln2sin 2 2 4

sin2

x zdx dz z xI tg C tg Cz

x dx dz

c)

2

22

2

2

2 21 2 22 2 1 111 1

1 2

dt dt tI dt C Ct xt t tt tgt

.


2/8


d)

2 22 2

2 2

1 2

2 1 1 4 1 155 3

1 1 2 4

12 1

2 2 2 1 22 2

15 15 15 15 15 15

2

dt dt dt I

t t t t tt

t t

xt tg

t

arctg C arctg C arctg c

2. Zadatak:Izraunati integrale:

a)3 5 cos

dxI

x

b)2 sin

2 cos

xI dx

x

c)sin

1 sin

xI dx

x

d)

1

5 4sin 3cosI dx

x x

Rjeenje:

a)2

222 1 2 11 2ln ln

2 4 4 2 41 23 5 221

xtg

dt ttI dt C Cxt tt tg

t

b)

22

2 2 2 2

2

22

2 11 41 1 1 3

21

t

dt t t tI dtt t t t

t

2

2 22 21 1 31 3t t At B Ct D

t tt t , 1 1, 0, , 1

2 2A B C D

2

2 2 2 2

22 2

2

2 2

2

2 2 4 2ln 1 4

1 3 3 3

4 3 4ln 1 ln 3 ln

13 3 3 3

342 2ln3 31

2

t t t dt I dt dt t dt

t t t t

t t tt t arctg C arctg C

t

x xtg tg

arctg C x

tg

c)1

cosI x tgx C

x

d)1

22

I Cx

tg

Ako je podintegralna funkcija parna2 2(sin ,cos )n mf x x dx integrali se rjeavaju smjenom


3/8


22 2

2 2 2

1cos , sin ,

1 1 1

tgx t

t dtx x dx

t t t

3.

Zadatak:Izraunati integrale:

a)21 3sin

dxI

x

b) 2 24cos 9sindx

Ix x

c) 2 24 3cos 5sin

dxI

x x

d)2 2sin 3sin cos cos

dxI

x x x x

Rjeenje:

a)

2222

2

1 12

11 4 4 231 1

41

dt dt dt I arctg tgx C

tt ttt

b)2

2 22

2 2

1 1 3 3 1 3141 4 9 9 9 2 2 6 2

4 991 1

dt

dt dt t tgxtI arctg C arctg Ct t

tt t

c)

2

2 22

2 2

1 1 31 313 5 9 1 9 9 1

491 1

1 33

dt

dt dt tI arctg t Ct t

tt t

arctg tgx C

d)

2

2 22

22

cos

sin sin 3 1 3 133 1 cos

cos cos 2 4

3 131 1 2 3 132 2ln ln13 3 13 13 2 3 13

2 2

dx tgx t dt dt xI dx

x x t tdttx

x x

ttgx

C Ctgx

t

4. Zadatak:Izraunati integral:

4cos

dxI

x

Rjeenje:


4/8


2

2 3 3

24

2

1 11 1cos 3 31

1

dt

dx tI t dt t t C tgx tg x Cx

t

Ako je podintegralna funkcija neparna 2 1 2 1(sin ,cos )n mf x x dx integrali se rjeavaju smjenomsinx t cosx t


)

3

2

sin

1 cos

xI dx

x

)3

2

cos

4sin 1

xI dx

x

)

2 22 2 2

1 cos sin cos 1 21

sin1 cos 1 1

2 cos 2 cos

x x x t tI dx dt dt

xdx dtx t t

t arctgt C x arctg x C

)1 3 2sin 1

sin ln4 16 2sin 1

xI x C

x

Ako je podintegralna funkcija oblika f tgx dx integrali se rjeavaju smjenom

tgx t , x arctgt 21

dtdx

t


)

1 2

dxI

tg x

) 2I tg x dx ) 3I ctg x dx ) 3

I tgx ctgx dx

Rjeenje:

)

22

2

1

2 11 ln 2 1 ln cos1 2 5 5 51 2 1

dt xtI dt tg x x Ct t t

)

2

2 2

11

1 1

tI dt dt t arctgt C tgx x C

t t

)21 ln sin

2I ctg x x C

) 2 21

2ln2

I tg x ctg x tgx C

7. Zadatak: Izraunati integral sin ( 2)nnS xdx n .Rjeenje:

1

sin sin

n

nS xdx

1 2sin ( 1)sin cos

sin cos

n nu x du n x xdx

dv xdx v x


5/8


1

1 2

1 2 2

1 2 2

1 2

1

sin sin

cos sin ( cos )( 1) sin cos

cos sin ( 1) cos sin

cos sin ( 1) (1 sin ) sin

cos sin ( 1) sin ( 1) sin

cos sin (

n

n

n n

n n

n n

n n n

n

S xdx

x x x n x xdx

x x n x xdx

x x n x xdx

x x n xdx n xdx

x x

2

1

2

1

2

1

2

1) ( 1)

( 1) cos sin ( 1)

(1 1) cos sin ( 1)

1 1sin cos

n n

n

n n n

n

n n

n

n n

n S n S

S n S x x n S

n S x x n S

nS x x S

n n

8. Zadatak: Izraunati integralsin cos

dx

x x .Reenje:

2 2

2 2

sin cos

sin cos sin cos

sin cos

sin cos sin cos

sin cos

cos sin

sinln cos ln sin ln

cos

ln

dx x xdx

x x x x

x xdx dx

x x x x

x x

dx dx tgxdx ctgx dxx x

xx x C C

x

tgx C

9. Zadatak: Izraunati integral4cos xdx .

Rjeenje:

2

4

1 coscos

2 2

1 coscos

2 2


6/8


2

4 2

2

2

1 cos 2 1cos (1 cos 2 )

2 4

1(1 2cos2 cos 2 )

4

1 1 1cos 2 cos 24 2 4

xxdx dx x dx

x x dx

dx xdx xdx

smjena: 2 (2 ) 22

dtx t x dx dt dx dt dx

1 1 1 1 1 cos 4cos 2 (2 )

4 2 2 4 2

1 1 1 1 1sin 2 cos 4 (4 )

4 4 8 8 4

1 1 1 1sin 2 sin 4

4 4 8 32

3 1 1sin 2 sin 4

8 4 32

xx xd x dx

x x dx xd x

x x x x C

x x x C

10. Zadatak: Izraunati integral2 24cos 9sin

dx

x x Rjeenje:

22 22 2

2

2 22

2

2 2

cos4cos 9sin (4cos 9sin )cos

4cos 9sin coscos

(4 9 ) cos

dx dx

xx x x xx

dx

x x xx

dx

tg x x

smjena:2

1

costgx t dx dt

x

2 2 24 9 2 (3 )

dt dt

t t

smjena: 3 33

dst s dt ds dt

2 2 2 2 2 2

13

2 (3 ) 2 3 2

1 1 1 3arctg arctg

3 2 2 6 2

1 3arctg

6 2

dsdt ds

t s s

s tC C

tgxC

tabliniintegral2 2

1arctg 0

dx xa

x a a a


7/8


11. Zadatak: Izraunati integral2

cos

2 sin

xdx

x .

Rjeenje:

smjena: sin cosx t x dx dt

2 2 2 2

cos

2 sin 2 ( 2)

sinarcsin arcsin

2 2

x dt dtdx

x t t

t xC C

tabliniintegral2 2

arcsindx x

Caa x

12.

Zadatak: Izraunati integral

arctg

(1 )

x

dxx x .Rjeenje:

smjena:1

22

x t dx dt dx tdtx

2 2

arctg arctg arctg2 2

(1 ) 1(1 )

x t tdx tdt dt

t t tx x

smena:2

1arctg

1t s dt ds

t

2

2

2 2

arctg2 2 2

1 2

arctg arctg

t sdt sds C

t

t C x C

13. Zadatak: Reiti integral xdx

Isin1

.

Rjeenje:

2 2

2

2

2

2 2

2 2sin ,

2 1 1

2

1 221 sin 2 1

11

1 22 2

11 1

2 21

12

x t dttg t x dx

t t

dx dt t dttx t t

t

d tdtC

tt t

C Cxttg


8/8


14. Zadatak: Izraunati integral21 x dx

Rjeenje:

smjena: sin cos , 1x t dx tdt x

2 2

2

1 1 sin cos

1cos (1 cos 2 )

2

1 1 1 1cos 2 cos

2 2 2 2 2

x dx t tdt

tdt t dt

dsdt tdt dt s

smjena: 2 22

dst s dt ds dt

22

1 1 1 1cos sin

2 4 2 41 1 1

sin 2 arcsin 12 4 2 2

dt sds t s C

xt t C x x C

integracija nekih trig. funkcija

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