Tabela de Derivadas e Integrais

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TABELA DE DERIVADAS 
 
Funções simples Funções compostas 
 
 ( x n )’ = n . x n – 1 
2
'
1
 
1
xx






 
 
x
x
2
1
 
'

 
( e x )’ = e x 
( a x )’ = a x. ln a sendo a = constante 
( ln x )’ = 
x
1
 
( log a x )’ = 
ax ln
1
 
( sen x )’ = cos x 
( cos x )’ = – sen x 
( tg x )’ = sec 2 x 
( cotg x )’ = – cossec 2 x 
( sec x )’ = sec x . tg x 
( cossec x )’ = – cossec x . cotg x 
( arc sen x )’ = 
21
1
x
 
( arc cos x )’ = – 
21
1
x
 
( arc tg x )’ = 
21
1
x
 
( arc cotg x)’ = – 
21
1
x
 
( arc sec x )’ = 
1
1
2 xx
 
( arc cossec x )’ = – 
1
1
2 xx
 
 
( u n )’ = n . u n – 1. u’ 
2
'
' 
 
1
u
u
u






 
 
u
u
u
2
' 
 
'

 
( e u )’ = e u. u’ 
( a u )’ = a u. ln a . u’ 
( ln u )’ = 
u
u '
 
( log a u )’ = 
au
u
ln.
'
 
( sen u )’ = cos u . u’ 
( cos u )’ = – sen u . u’ 
( tg u )’ = sec 2 u . u’ 
( cotg u )’ = – cossec 2 u . u’ 
( sec u )’ = sec u . tg u . u’ 
( cossec u )’ = – cossec u . cotg u . u’ 
( arc sen u )’ = 
21
'
u
u

 
( arc cos u )’ = – 
21
'
u
u

 
( arc tg u )’ = 
21
'
u
u

 
( arc cotg u )’ = – 
21
'
u
u

 
( arc sec u )’ = 
1
'
2 uu
u
 
( arc cossec u )’ = – 
1
'
2 uu
u
 
 
Propriedades: ( k . u )’ = k . u’ 
 ( u . v)’ = u’.v + u . v’ 
 
 
2
'
'.'.
v
u
v
vuvu 






 
TABELA DE INTEGRAIS 
 
 
 


c
n
u
duu
n
n
1
1 
  cuduu
||ln
1
 
  caa
dua uu
ln
1
 
  cedxe
 x x 1 

 
  cxxsen cos
1
 
 
  cxsenx 
1
 cos 
 
  cuduusen cos 
 
  cusenduu cos
 
  cutgduu sec
2
 
  cuguec cotcos
2
 
cuduutgu  sec sec
 
cuecduuguec  cos cot cos
 
cuduutg  | cos|ln 
 
cusenduug  | |ln cot
 
cutguduu | sec |ln sec 
 
cuguduuec | cot secco |ln cos 
 
  cxfdxxf
xf
|)(|ln
)(
)(' 
 
 
c
a
u
tgarc
a
du
ua
 
11
22
 
 



c
au
au
a
du
ua
 ln
2
11
22
 
 



c
au
au
a
du
au
 ln
2
11
22
 
 
 
 

c
a
u
senarcdu
ua
 
1
22
 
 

cauudu
au
||ln
1 22
22
 
 

c
a
u
arc
a
du
auu
sec 
11
22
 
 





 dxxdxxsen 2cos
2
1
2
1
 2
 
 





 dxxdxx 2cos
2
1
2
1
 cos2
 

 

 dxxsen
n
n
xxsen
n
dxxsen nnn 
1
cos . 
1
 21
 

 

 dxx
n
n
xsenx
n
dxx nnn cos
1
 . cos 
1
 cos 21
 

 




 dxx
n
n
xtgx
n
dxx nnn sec
1
2
 . sec 
1
1
 sec 22
 

 




 dxxec
n
n
xx
n
dxxec nnn cos
1
2
 cotg . secco 
1
1
 cos 22
 
 
Integração por partes: 
  dvu
 = u . v –
  duv
 
 
Substituição trigonométrica: 
22 xa 
 

 x = a . sen θ 
22 xa 
 

 x = a . tg θ 
22 ax 
 

 x = a . sec θ 
Originando as integrais: 
  ca
x
arcsen
a
xa
x
dxxa
2
²
2
2222
 
  cxax
a
xa
x
dxxa 222222 ln
2
²
2
 
  caxx
a
ax
x
dxax 222222 ln
2
²
2