NEODREĐENI INTEGRAL

[Pages:5]NEODREENI INTEGRAL

Funkcija F(x) je primitivna funkcija funkcije f(x) ako je F'(x)=f(x). Ako je F primitivna za f onda je i svaka f-ja F(x)+C ( C je konstanta) takoe primitivna funkcija f-je f(x). Skup svih primitivnih f-ja funkcije f naziva se neodreeni integral funkcije f i obelezava sa:

f (x)dx

TABLICA INTEGRALA:

1. 0dx C

Neodreeni integral predstavlja povrsinu izmeu f(x) i x-ose

2.

xndx

xn1

+C

n1

3.

1 x

dx

ln

x

C

4.

ax dx

ax lna

C

5. exdx ex C

y

f (x)dx

6. cos xdx sinx C

x

7. sinxdx cosx C

8.

dx cos2

x

tgx

C

9.

dx s in2

x

ctgx

C

10.

dx arcsinx C arccosx C 1 x2

dx

11. 1 x2

arctgx C

arcctgx

C

OSOBINE NEODREENOG INTEGRALA:

1. Cf(x)dx C f(x)dx

2. f(x) g(x)dx f(x)dx g(x)dx

Primeri:

1. 3x4

5 x2

3 x dx

3

x

4dx

5

x

2dx

x1/

3dx

3

x5 5

5 x1 1

x4/3 4/3

C

3x5 5

5 x

33 x4 4

C

2.

1

2 x2

3 1 x2

5ex

s

4 in2

x

dx

2

1

1 x

2

dx

3

2arctgx 3arcsinx 5ex 4ctgx C

1 1 x2

dx

5

exdx

4

1 s in2

x

dx

Pomou tablice integrala i osobina izracunati sledee integrale

1.

x2

2x

1 x

dx

2.

6x 4

8x7 / 2 x

5x

2

dx

3. x4

x3 xx

x 3dx

4. au1

au u3

du

5. 3y e3y dy

6.

2x

1 5x 1 10x

dx

7.

s

cos 2x in2 x cos2

x

dx

10.

1 x2

x 1 x2

dx

8.

s

in2

dx x cos2

x

11.

1

s

s in3 in2 x

x

dx

9.

4

cos

y

9

5 9z2

dz

12.

s

in

x 2

cos

x 2

2

dx

INTEGRACIJA METODOM SMENE

Neka je x=g(t), gde je t nova promenljiva i g'(t) 0.Tada je dx=g'(t)dt pa je f(x)dx f(g(t))g'(t)dt

Smena se uvodi tako da novi integral bude pogodan za integraciju.

Primeri:

1.

5 2x9dx

smena: 5 2x t - 2dx dt dx - dt 2

t9

dt 2

1 2

t9dt

1 2

t10 10

t10

20

5 2x10

20

C

2.

xdx 1 x2

2

smena: 1 x2 t 2xdx dt dx

dt 2x

x dt

2x t2

1 dt 2 t2

1 2

t

2dt

1 t1 1 2 1 2t

2(1

1

x

2

)

C

Metodom smene resiti sledee integrale

13.

dz 3z

2

17.

x

2

dx a2

21.

ex 1 e2x

dx

14.

xdx 1 x2

18.

2

dx 3x

2

22.

x(1

dx ln2

x)

25. dz 9 16z2

29.

ex ex

dx 1

33. esinx cos xdx

37.

x2

dx 4x

13

26. cosxdx a2 sin2 x

30. ex2 xdx

1

34.

1 x2

e

x

dx

38.

2x2

dx 2x

1

15.

x2dx x3 1

19.

cos xdx 4 sin2 x

23. dx 25 9x2

27. exdx 1 e2x

31.

e2xdx 1 3e2x

35.

x

dx ln x lnln x

39.

dx

2 3x x2

16.

2x x2

x

1

3

dx

20.

s inxdx a2 cos2

x

24. 7dx 3 5x2

28. dx x x2 1

32. e x dx x

36. cos x dx x

40.

dx

2x2 6x 5

PARCIJALNA INTEGRACIJA

udv uv vdu

Primeri:

1. ln xdx

lnx u 1 dx du x

dx dv dx v x

v

x

ln

x

x

1 x

dx

x lnx dx

x lnx x C

2.

x2

2x 5 exdx

x2 2x 5 u (2x 2)dx du exdx dv v exdx v ex

ex

x2

2x 5

2x 2exdx

ex x2 2x 5 2 x 1exdx ex x2 2x 5 2I ex x2 2x 5 2 ex (x 1) ex C

ex (x2 2x 5 2(x 1) 2) C ex (x2 5) C

I

x

1e

xdx

x e

1 xdx

u

dv

dx

v

du e

x

ex (x 1) exdx ex (x 1) ex

Parcijalnom integracijom resiti sledee integrale:

41. xe xdx

42. x2 cos xdx

43. arctgxdx

45. x2 5x 6 cos2xdx

49. ex cos xdx 53. xarctgxdx

46.

xdx cos2

x

50. x3 cosxdx

54. x3ex2 dx

47.

lnx x3

dx

51. x2 a2dx

55.

x cos x sin2 x

dx

44.

x ex

dx

48. ex sinxdx

52. x2 a2dx 56. ex sin2 xdx

INTEGRACIJA RACIONALNIH FUNKCIJA

Primeri:

1.

(x

xdx 1)(x2

1)

*

*

*

(x

x 1)(x2

1)

A x 1

Bx x2

C 1

x A(x2 1) (Bx C)(x 1)

x 1 x2 1 => x Ax 2 A Bx2 Bx Cx C

x x2(A B) x(B C) A C

=> A+B=0 => A=-B => -B+C=1

-B+C=1

-B-C=0 + => -2B=1=>B=-1/2,A=1/2,C=1/2

A-C=0

(x

x 1)(x2

1)

1 2

x

1 1

1 2

x 1 x2 1

***=

1 2

dx x 1

1 2

x 1 x2 1

1 lnx 2

1

1 2

xdx

x2 1

dx

x2

1

1 lnx 2

1

1 2

1 lnx2 2

1

arctgx

C

smena: x - 1 t

smena: x 2 1 t

2.

(x

dx 1)2 (x

2)

*

*

*

(x

1 1)2(x

2)

A x 1

(x

B 1)2

C x2

x 12x 2

=> 1=A(x-1)(x-2)+B(x-2)+C(x-1)2 1= AX2-3Ax+2A+Bx-2B+Cx2-2Cx+C 1=x2(A+C)+x(-3A+B-2C)+2A-2B+C

A+C=0

-3A+B-2C=0

2A-2B+C=1

A=-C

-3(-C)+B-2C=0

2(-C)-2B+C=1

B+C=0

-2B-C=1 +=> -B=1=> B=-1 ; C=1; A= -1

(x

1 1)2(x

2)

1 x 1

(x

1 1)2

x

1 2

* **

(x

1 1)2(x

2)

dx

x

1

dx 1

(x

1 1)2

dx

1 dx lnx 1 1 lnx 2 C

x2

x 1

lnx 1 x 2 1 C x 1

Odrediti integrale racionalnih f-ja:

57.

x2 x3 2x2

dx

58.

x3

x2 16x x2 4x 3

16

dx

x2 2

59. (x

2)(x

1)3

dx

dx 60. (x 1)2(x 2)

dx 61. x4 x2

62.

x x3

5

1

dx

dx 63. (x2 3)(x2 2)

dx 64. x3 1

65.

x3

3x x2

7 4x

4

dx

Koristei smenu t tg x 2

s inx

2t 1 t2

;cos x

1 1

t2 t2

;dx

2dt 1 t2

odrediti

integrale:

dx

66.

1 sinx cosx

dx

67.

5 4sinx 3cosx

68. 1 sinx cosx dx 1 sinx cosx

RESITI SLEDEE INTEGRALE

69.

1 x dx 1 x

72. xln x2 1dx

70. x3 lnxdx

73.

x5 x3

2 1

dx

71.

exdx

1 ex e2x

74. x sinx cosxdx

75. cos4 xdx

78. xarctgx2dx

81.

e4x

5ex 3e2x

dx 4

84.

1

s

dx inx

cos

x

87. sinx cos xdx 2 sin4 x

90. dx

2x x2

76.

1 lnx dx x

79. sin(lnx)dx

82. x2arctg3xdx

85.

3

dx 5 cos

x

88.

2

s

in2

x

dx 3

cos2

x

91.

x3 x 1 x2 2 2

77. x2 x lnx 1dx

80.

ln(x 1) lnx x(x 1)

dx

83.

1

s inx sinx

dx

86. dx x 1 x2

89.

x

2

dx a2

92. ex cos2 xdx

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