Tabela_Integrais_Derivadas

Prévia do material em texto

1
 
TABELA DERIVADAS E INTEGRAIS 
 
TABELA DE DERIVADAS 
 
Sejam u e v, funções deriváveis de x, e sejam a, c e n constantes. 
 
2
2
2
1. ' 0
2. ' 1
1 13. '
4. ( 0, 1) ' ln '
5. ' '
'6. log ( 0, 1) '
ln
'7. ln '
8. ' cos '
9. cos ' '
10. ' sec '
11. cot ' cosec '
12.
u u
u u
u
a
y c c y
y x y
y y
x x
y a a a y a a u
y e y e u
uy a a y
u a
uy u y
u
y senu y u u
y u y senu u
y tg u y u u
y g u y u u
y
= ∈ ℜ =
= =
= = −
= > ≠ = ⋅ ⋅
= = ⋅
= > ≠ =
⋅
= =
= = ⋅
= = − ⋅
= = ⋅
= = − ⋅
2
sec ' sec '
12. sec ' sec '
13. cosec ' cosec cot '
'14. '
1
u y u tg u u
y u y u tg u u
y u y u g u u
uy arc senu y
u
= = ⋅ ⋅
= = ⋅ ⋅
= = − ⋅ ⋅
= =
−
 
2
2
2
2
2
1
1
'15. arccos '
1
'16. '
1
'17. cot '
1
'18. sec , | | 1 ' , | | 1| | 1
'19. osec , | | 1 ' ,| | 1| | 1
20. ( 0) ' '
21. ( 0) ' ' ln '
22.
n n
v v v
uy u y
u
uy arc tg u y
u
uy arc g u y
u
uy arc u u y u
u u
uy arc c u u y u
u u
y u n y nu u
y u u y v u u u u v
y u v
−
−
= = −
−
= =
+
= = −
+
= ≥ = >
−
= ≥ = − >
−
= ≠ =
= > = +
= +
2
' ' '
23. ' '
24. ' ' '
' '25. '
y u v
y c u y c u
y u v y u v u v
u u v u vy y
v v
= +
= ⋅ = ⋅
= ⋅ = ⋅ + ⋅
−
= =
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Docente: Ivana Barreto Matos Disciplina: Funções Multivariáveis e Analíticas 
 2
 
TABELA DE INTEGRAIS 
 
(1) ∫ += cxdx 13) ∫ +=
−
cx
x
dx
 arcsen 
1 
 
2
 
(2)
1
 , , 1lim
1
a
a
x
x
x dx c a a
a
+
→∞
= + ∈ℜ ≠ −
+∫
 
(14) ∫ +=
−
cxarc
xx
dx
 sec 
1 
 
2
 
(3)
 ln dx x c
x
= +∫ 
(15) cx c x tgx dx sec ln cos ln +=+−=∫ 
(4) ∫ += cedxe
xx
 
(16) cx cx gx dx seccos ln sen ln cot +−=+=∫ 
(5) ∫ ≠>+= 10 ln
1
 , a, aca
a
dxa xx (17) ctgxx dxx sec ln sec ++=∫ 
(6) ∫ +−= cxdxx cos sen (18) cgxx dxx cot seccos ln seccos +−=∫ 
(7) ∫ += c sen cos xdxx (19) 0 ) ( 1 22 ≠+=
+
∫ , aca
x
arctg
axa
dx
 
(8) ∫ += c sec 
2 tgxdxx (20) 0 ) ( arcsen
 
22
≠+=
−
∫ a, ca
x
 
xa
dx
 
 
(9) ∫ +−= c cot seccos 
2 gx dxx (21) 0 ) (sec1 
 
 
22
≠+=
−
∫ , aca
x
 arc
aaxx 
dx
 
(10) ∫ += cx tgx dxx sec sec (22) caxx 
ax
dx
 ln 
 
 
22
22
+±+=
±
∫ 
(11) ∫ +−= cxgx dxx seccos cotseccos (23) ∫ ++
−
=
−
c
ax
ax
 
aax
dx
 ln 
2
1
 22 
(12) ∫ +=
+
carctgx
x
dx
 
1
 2 
Obs.: c é uma constante real. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3
 
EXPRESSÕES TRIGONOMÉTRICAS 
 
1) 1 cos sen 22 =+ xx 2) xtgx 22 1 sec += 
3) xgx 22 cot 1 seccos += 4) ) 
2
 ( sen 2 cos 1 2 xx =− 
5) ) 
2
 ( cos 2 cos 1 2 xx =+ 6) 
2
 ) 2 ( cos 1 
 sen
2 x
x
−
= 
7) 
2
 ) 2 ( cos 1 
 cos
2 x
x
+
= 
 
8) x xx cos sen 2 ) 2 ( sen = 
 
9) xxx 22 sen cos ) 2 ( cos −= 10) 
xtg
tgx
x 2
 1
 2
 ) 2 ( sen
+
= 
11) 
xtg
xtg
x 2
2
 1
 1
 ) 2 ( cos
+
−
= 
12) 
 ) 
2
 ( 1
 ) 
2
 ( 2
 sen
2 xtg
x
tg
x
+
= 
13) 
 ) 
2
 ( 1
 ) 
2
 ( 1
 cos
2
2
x
tg
x
tg
x
+
−
= 
14) 
 1 
 sen 2
2
2
xtg
xtg
x
+
= 
15) 
 1 
1
 cos 2
2
xtg
x
+
= 
 
16) } ] ) ( [ cos ] ) ( [ cos { 
2
1
 ) (cos ) (cos xnmxnmnx . mx −++= 
17) } ] ) ( [ sen ] ) ( [ sen { 
2
1
 ) (cos ) (sen xnmxnmnx . mx −++= 
18) } ] ) ( [ cos ] ) ( [ cos { 
2
1
 ) (sen ) (sen xnmxnmnx . mx +−−=