Tabel - Derivate Si Integrale 2p3r1l

Nr. 1 2 3

Derivate

Nr.

c 0 x'  1 '

x   nx n '

n 1

4

 x  21x

5

1 1    2 x  x

1

 dx  x  C

2

 xdx 

'

3

'

7

e   e a   a

8

ln x '  1

9

log a x ' 

6

10 11 12 13 14 15 16 17 18 19 20

x '

4 5

x

x '

x

ln a

x

1 x ln a arctg x '  21 x 1 arcctg x '   21 x 1 arcsin x '  1 2 1 x arccos x '   1 2 1 x

7

 x dx  ln x  C

10 11 12

'

13

tg x ' 

x x a

2

2

2

14

 x x a   x x a    a x x

15

'

 a2   a2  x2

2

16

2

17

'

2

2

18

'

2

2

ax C ln a

x  a dx 

9

1 cos 2 x ctg x '   12 sin x

x2 C 2 x n 1 n  x dx  n  1  C 2  x dx  3 x x  C x x  e dx  e  C

6

8

sin x   cos x cos x '   sin x

Integrale nedefinite

1

1 1 xa dx  ln C 2 2a x  a a 1  x 2  1 dx  arctg x  C 1 1 x  x 2  a 2 dx  a arctg a  C 1 2 2  x 2  a 2 dx  ln x  x  a  C 1 2 2  x 2  a 2 dx  ln x  x  a  C 1  1  x 2 dx  arcsin x  C 1 x  a 2  x 2 dx  arcsin a  C

x

2



 sinx dx   cos x  C  cos x dx  sinx  C  tgxdx   ln cos x  C  ctgxdx  ln sinx  C 1

19

 cos

20

 sin

21



22



23



2

x

1 2

x x

dx  tg x  C dx   ctg x  C

x a x

2

x a x

2

a x

2

2

2

2

dx  x 2  a 2  C dx  x 2  a 2  C dx   a 2  x 2  C



Nr. crt. 1 2 3

Operaţii

 f  g   f   g   f  g   f   g  f  g  cf   c  f  

4

5

f  f   g  f  g    g g2

  f ( x)  g ( x)dx   f ( x)dx   g ( x)dx

Formule Derivarea funcţiilor compuse  f (u ) '  f ' (u )  u ' Derivata fucţiei inverse  f 1 ' ( y)  '1 , unde y  f ( x) f ( x) Formula Leibniz-Newton b

 f ( x)dx  F ( x)|

b a

 F (b)  F (a ), F o primitiva f

a

Integrarea prin părţi 6

   f ( x)dx   f ( x)dx

b

 a

7

  f ( x)  g ( x)dx   f ( x)dx   g ( x)dx

b

a

a

Prima schimbare de variabilă b

 a

b

f ( x) g ' ( x)dx  f ( x) g ( x) |   f ' ( x) g ( x)dx f  ( x)    ' ( x)dx 

 (b )

 f (t )dt

 (a)