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tablicni integrali

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1. f ( x ) g ( x ) dx = f ( x ) dx g ( x ) dx

2. kf ( x )dx = k f ( x ) dx 3. f ( x )dx = f ( x ) d ( x a ) 4.

k = const a = const a0, a = const

1 f ( x )dx = f ( x ) d ( ax ) a

5. dF ( x ) = F ( x ) + C

1. 2. 3. 4. 5. dx = x + Cx p +1 x p dx = +C p +1 dx x = ln x + C e x dx = e x + C

p -1

ax a x dx = +C ln ( a )

6. sin xdx = cos x + C 7. cos xdx = sin x + C

8. 9. 10. 11. 12.1 cos2 x dx = tg x + C 1 sin 2 x dx = cotg x + C 1 1 x x 2 + a 2 dx = a arctg a + C

x dx = arcsin + C a2 x2 a 1 1 xa +C ln dx = x2 a2 2a x + a 1

a>0

13.

1 x2 a2

dx = ln x + x 2 a 2 + C

:

x x3 2 x dx = +C (2) 1 x dx = + C p +1 3 1 x 2 1 3 +C = 2 +C 2 3 dx = x dx = x 2x 2 7 2 3 3 3 5 x 4 x + 2 3 2 dx = x x 1 1 7 3 = 5 x dx 4 x dx + 2 2 dx 3 2 dx = x x3 2p p +1

x x x x 5x 2 4 =5 4 +2 3 +C = x 9x + C 1 8 4 8 x 1 3

8

4

1

+1 3

8

1 3

4

x +C dx = arcsin 5 5 x2

1

(11)

x dx = arcsin + C a a2 x2 1

5 1 1 3 + x 2 + x 2 + 3 dx =

( 3)

12

+x

2

dx +

1 x2 + 3

dx =

1 x arctg = + ln x + x 2 + 3 + C 3 3

67 2

7 2 x x + x 3 dx = 9 2

1 2x 3x = x dx + 2 dx 3x dx = + 2 ln x +C x 9 ln 3

7x2 x2 1 + 1 1 x2 1 1 x 2 dx = 1 x 2 dx = 1 x 2 dx x 2 1 dx =

1 1 x 1 1 x +1 = dx 2 dx = x ln + C = x + ln +C x 1 2 x +1 2 x 1

81 x2 x2 + 2 2 x 2 + 2 dx = x 2 + 2 dx = dx 2 x 2 + 2 dx =1 x arctg = x2 +C 2 2

92 x 5x + 3 x 2 1 dx =4 2

2 x 2 3)( x 2 1) ( x 12

dx =

2 3 = 2 x dx 3 dx = x 3 x + C 32

10

2 + x3 x5 1 x2

dx =

2 1 x2

dx +

x3 x5 1 x2

dx =3 2

= 2 arcsin x + C +

x3 1 x 2 1 x2

dx = 2 arcsin x + C + x dx =

2 5 = 2 arcsin x + x 2 + C 5

11sin 2 x 1 cos 2 x dx = dx = tg 2 xdx = 2 2 cos x cos x 1 1 dx dx = tg x x + C = 1 dx = 2 2 cos x cos x

122 + 3x 2 2 x2 x 4 + x2 dx = x 2 ( x 2 + 1) dx + 3 x 2 ( x 2 + 1) dx =

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

2 2 = 2 dx 2 dx + 3arctg x + C = x x +1 2 2 = 2 arctg x + 3arctg x + C = + arctg x + C x x

13

1 f ( ax + b ) dx = f ( ax + b ) d ( ax + b ) a

1 1 x 3 dx = x 3 d ( x 3) = ln x 3 + C

14

( x + 7)

17

dx = ( x + 7 )

17

1 18 d ( x + 7) = ( x + 7) + C 18

15

1 1 1 dx = d ( 5 x ) = arcsin 5 x + C 2 2 5 1 (5x ) 5 1 25 x

1

161 1 1 1 9 + 4 x 2 dx = 32 + ( 2 x )2 dx = 2 32 + ( 2 x )2 d ( 2 x ) =11 2x = arctg + C 23 3

17

e

13 x2

1 3 x + 2 1 3 x + 2 dx = e d ( 3 x + 2 ) = e +C 3 3

181 cos 5 x sin 5 xdx = 5 sin 5 xd ( 5 x ) = 5 + C

19

1 5x 42

dx =

1 4 (5x 4) . 42

dx =

1 x 5 2 1 2 2

dx =

x 5 1 1 1 1 = dx = d = 2 2 2 2 5 x 5 x 5 1 1 2 2 1 5x2 x 5 = ln + 1 + C 2 4 5

202

1 + cos 2 x 1 11 cos xdx = 2 dx = 2 dx + 2 2 cos 2 xd ( 2 x ) = 1 1 = x + sin 2 x + C 2 4

21 1 1 1 1 cos2 5 x + x 2 4 dx = cos2 5 x dx + x 2 4 dx = 1 1 = d ( 5 x ) + ln x + x 2 4 + C = 5 cos 2 5 x 1 2 = tg 5 x + ln x + x 4 + C 5

221 cos 2 x 1 cos 2 x sin xdx = 2 dx = 2 2 dx = 1 cos 2 x 1 11 = dx dx = x + C cos 2 xd ( 2 x ) = 2 2 2 22 1 1 = x sin 2 x + C 2 42

23

1 = 5

1 1 dx = x+ 2 x3 x+ 2 x3

x+ 2 + x3 dx = x+ 2 + x3

(

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

)

3 3 2 2 + ( x 3) 2 + C = ( x + 2 ) 15

( 3 x + 5 )7 + e 2 x dx = ( 3 x + 5 )7 dx + e 2 x dx = 24

1 1 2x 1 ( 3x + 5) e2 x 7 = ( 3x + 5) d ( 3x + 5) + e d ( 2 x ) = + +C 3 2 3 8 28

f ( x ) g ( x ) dx = g ( x ) d ( f ( x ) dx ) 25

cos x3

sin 2 x

dx = sin

2 3

( sin x ) xd sin x =

2 +1 3

2 +1 3

+ C = 3 3 sin x + C

26ln 2 x 1 2 dx = ln x dx = ln 2 xd ( ln x ) = x x 3 2 ( ln x ) + C = ( ln x ) d ( ln x ) = 3

27 x 1 x2 1 1 x2 1 dx = d = (1 x 2 ) 2 dx 2 = 1 x2 2 2

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

1 +1 2 2

= 1 x2 + C

281 1 1 1 (1 + x 2 ) arctg x dx = (1 + x 2 ) . arctg x dx = arctg x d ( arctg x ) =

= ln arctg x + C5x 3 x 7 + 2 x 2 dx =5 7 + 2 x 2 dx 3

29

( 7) +(2

1

2x

)

2

dx =

51 1 3 2 = 7 + 2 x2 d ( 2 x + 7 ) 2 22

( 7) +(2

1

2x

)

2

d

(

2x =

)

5 3 2x 2 = ln ( 7 + 2 x ) +C arctg 4 14 7

x 2x + 3 1 3 + 4 x 2 dx =2 3 + 4 x 2 dx + 3 3 + 4 x 2 dx =1 x2 = 2 d + 3 2 3 + 4x 2

30

( 3)

12

+ ( 2x)

2

dx =12

21 1 3 2 = 3 + 4 x 2 d ( 4 x + 3) + 2 24

( 3)

+ ( 2x)

2

d ( 2x) =

1 3 1 2x 2 arctg = ln 3 + 4 x + +C = 4 2 3 3

1 3 2x 2 arctg = ln 3 + 4 x + +C 4 2 3

31

3x + 4

5 x2

dx =3

x 5 x2

dx + 4

1

( )5

2

dx = x2

3 1 x 2 = +C = d ( x ) + 4 arcsin 2 5 5 x2 x 3 1 2 = +C = d ( x + 5 ) + 4 arcsin 2 5 5 x21 x 3 2 2 d 5 x 2 + 4 arcsin = (5 x ) +C = ( ) 2 5 x 2 = 3 5 x + 4 arcsin +C 5

32

sin x 1 tg xdx = cos x dx = cos xd ( cos x ) = ln cos x + C

33 1 sin x dx =

1 1 1 x x dx = d = d = x x x x 2 x x x cos 2 2 2sin cos sin cos 2 2 2 2 sin cos 2 2 cos x 2 1 1 x x x = = = ln tg + C d d tg 2 x 2 2 x x 2 tg tg cos 2 2 2

34

1 + tg x 1 1 4 2 dx = d x = cos x sin x 2 2 tg x 2 4 2

(

)

( ((

)) d x = 2 )2 2

sin x 4 2 1+ cos x 4 2 = tg x 4 2

(

( (

)

) )

1 cos x x 4 2 d == 4 2 tg x 4 2

2

( ((

)) d x = 4 2 )sin x =

=

tg x 4 2

(

1

)

x x d tg = ln tg + C 4 2 4 2

2 tg x 1 + tg x

(

2 2

)

2

u ( x ) d ( v ( x ) ) =u ( x ) v ( x ) v ( x ) d ( u ( x ) ) 35

x arccos x 1 x2

dx =

ex 1 x2 dx =

x 1 x2

d.

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

= arcsin xd 1 x 2 = . = 1 x 2 arcsin x + 1 x 2 d arcsin x = = 1 x arcsin x + 1 x2 2

1 1 x2

dx =

= 1 x 2 arcsin x + dx = 1 x 2 arcsin x + x + C

361 1 x 2x 1 2x 2x 2x xe dx = 2 xe d 2 x = 2 xd ( e ) = 2 e 2 e dx =2x

x 2x 1 1 2x x 2x 1 2x 1 2x x = e e +C = e e +C = e +C 2 22 2 4 2 4

37x arctg xdx =x arctg x xd arctg x =x arctg x 1 + x 2 dx = 1 1 1 2 = x arctg x d (1 + x ) =x arctg x ln (1 + x 2 ) + C 2 1 + x2 2

381 1 4 1 4 4 x arctg xdx = 4 arctg xdx = 4 x arctg x 4 x d arctg x = 1 4 1 1 4 1 x4 1 + 1 x4 dx = x arctg x dx = = x arctg x 2 2 4 4 1+ x 4 4 1+ x ( x 2 1)( x 2 + 1) dx 1 1 dx = 1 4 1 = x arctg x 4 4 1 + x2 4 1 + x2 1 4 1 1 2 = x arctg x ( x 1) dx arctg x + C = 4 4 4 1 4 1 2 1 1 = x arctg x x dx + dx arctg x + C = 4 4 4 4 1 4 1 3 1 1 = x arctg x x + x arctg x + C 4 12 4 43

391 1 2 1 2 2 2 2 x arctg xdx = arctg xdx = x arctg x x d arctg 2 x = 2 2 2 1 2 2 1 2 x2 x2 + 1 1 arctg xdx = x arctg 2 x arctg xdx = = x arctg 2 x 2 2 2 2 1+ x 2 1+ x 1 2 1 2 arctg xdx = = x arctg x arctg xdx + 2 2 1+ x 1 2 = x arctg 2 x x arctg x + xd arctg x + arctg xd arctg x = 2 1 2 x 1 2 dx + arctg 2 x + C = = x arctg x x arctg x + 2 1 + x2 2 1 2 1 1 1 2 2 = x arctg x x arctg x + d ( x + 1) + arctg 2 x + C = 2 2 1 + x2 2 1 2 1 1 2 2 = x arctg x x arctg x + ln x + 1 + arctg 2 x + C 2 2 22

40arccos x 1 1 1 x 2 dx = arccos xd x = = x arccos x x d arccos x = 1 1 1 1 = arccos x dx = arccos x dx = x x x 1 x2 x2 x (1 x 2 ) 2 x 1 1 1 1 1 arccos x dx = arccos x + d = 2 x x x 1 2 1 x 1 2 1 x x 1 1 1 = arccos x + ln + 2 1 + C x x x

41

x ln x + 1 + x 2 1 + x2

(

) dx2

e

x 1+ x

d.

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

ln x + 1 + x 2 d 1 + x 2 =

(

= 1 + x 2 ln x + 1 + x 2 1 + x 2 d ln x + 1 + x 2 = = 1+ x2 2

( ln ( x +

)

) 1+ x )

(

)

1 1 1+ x 2 x dx = 1 + 2 2 x + 1+ x 2 1+ x 2

1

= 1 + x ln x + 1 + x2

( (

2

)

x 1 + 1 + x2 x + 1 + x2 1 + x2 1 + x2 1 + x2 dx

dx = dx =

= 1 + x ln x + 1 + x2

2

) x+

x x + 1 + x2

x 1 + x2 1 x2 dx = dx = dx = 2 2 2 2 2 x 1 x x + 1+ x x + 1+ x x 1+ x = x 1 + x 2 1 x 2 dx = x 1 + x 2 dx dx x 2 dx

1 + x2

1 + x2

x 1 + x2

(

)

x2 x 1 + x2 dx = dx = 2 dx = 2 2 2 2 x 1 x x + 1+ x x + 1+ x x 1+ x x x = x 2 x 1 + x 2 dx = x 2 dx x 1 + x 2 dx

x 1 + x2

(

)

= 1 + x 2 ln x + 1 + x 2 + x 1 + x 2 dx dx x 2 dx + x 2 dx x 1 + x 2 dx = = 1 + x 2 ln x + 1 + x 2 dx = = 1 + x22

(

)

( ln ( x +

) 1+ x ) x + C

Mathematica: : 7 2 3 3 5 x 4 x + x 2 3 x 2 dx i 7 y j j5 x 4 x3 + 2 3 z x z z [email protected]_D := j j j 2 !!!!! z 3 2 z x k x {8 2 9 Hx2L23 4 5x x + x 8 x

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