Tabela de derivadas

TABELA BÁSICA DE DERIVADAS
Constante Potência

TABELA BÁSICA DE INTEGRAIS
Propriedades: n–1

( c )` = 0 ( x )` = n.x x x n n–1

( u ) ` = n.u

n

∫ n . f ( x ). dx
∫ [ f (x) +

= n .∫ f ( x ) + c

( n é constante )

.u`

g ( x )]. dx =

∫

f ( x ). dx +

∫ g ( x ). dx

( a ) ` = a .ln a
Exponencial

( au ) ` = u`.au. ln a ( eu ) ` = u`.eu

Método de Integração por Partes:

( ex ) ` = ex

∫
∫
a b

u . dv

= u .v − b ∫

v . du

Logarítmica

Neperiana

1 ou a x . ln a u` u (log a )` = ou u . ln a 1 (ln x )` = (ln u x (log x )` =

1 e . log a x u` e . log a u u` )` = u

Integral Definida:

f ( x ). dx = F ( x ) a ⇔ F (b ) − F (a )

∫ du = u + c

∫
∫

u n du =

u n +1 + c n +1

( n é constante ≠ – 1 )

(sen x)` = cos x (cos x)` = – sen x
Trigonométrica

(sen u)` = u`.cos u (cos u)` = – u`.sen u (tg u)` = u`.sec² u (cotg u)` = – u`.csc² u (sec u)` = u`.sec u . tg u (csc u)` = – u`.cscu.cotgu u` ( arcsen u )`= 1− u²
− ( u `) (arccos u )`= 1− u²

∫ ln du u u u . du = u . ln u − u + c
= ln u + c
= eu + c

∫
∫

(tg x)` = sec² x (cotg x)` = – csc² x (sec x)` = sec x . tg x (csc x)` = – cscx.cotg x 1 ( arcsen x )`= 1 − x²
(arccos x )`= −1 1 − x²

1 . dx x e kx

= ln x + c
= e kx ∫e

du

. dx

k

+ c

∫ senu .du = − cos u + c ∫ cos u.du = senu + c ∫ tgu .du = ln sec u + c
∫ cot gu.du = ln senu + c ∫ sec u .du ∫ csc u .du ∫ tg ² u .du

∫ senkx .dx

=−

cos kx +c k senkx + c

∫ cos kx .dx = k ∫ tgu .du = − ln cos

u +c

Trigonométrica Inversa

( arctg x )`=

1 1 + x²
−1 1 + x²

( arctg u )`=

u` 1+ u²
− ( u `) 1+ u²

= ln sec u + tgu + c

∫ sec ² u .du = tgu + c
= ln csc u − cot gu + c

( arc cot g x )`=

( arc cot g u )`=

( arc sec x )`=

1 x. x ² − 1

( arc sec u )`=

u` u. u ² − 1

∫ csc ² u .du = − cot gu + c
= tgu − u + c = sec u + c

−1 ( arc csc x )`= x. x ² − 1
Soma

− ( u `) ( arc csc u )`= u. u ² − 1