Tabela derivadas e integrais

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UNIVERSIDADE FEDERAL DO ABC
Tabela de Derivadas, Integrais e Identidades Trigonome´tricas
Derivadas
Regras de Derivac¸a˜o
• (cf(x)) 0 = cf 0(x)
• Derivada da Soma
(f(x) + g(x)) 0 = f 0(x) + g 0(x)
• Derivada do Produto
(f(x)g(x)) 0 = f 0(x)g(x) + f(x)g 0(x)
• Derivada do Quociente✓
f(x)
g(x)
◆ 0
=
f 0(x)g(x)- f(x)g 0(x)
g(x)2
• Regra da Cadeia
(f(g(x)) 0 = (f 0(g(x))g 0(x)
Func¸o˜es Simples
• ddxc = 0
• ddxx = 1
• ddxcx = c
• ddxxc = cxc-1
• ddx
�
1
x
�
= ddx
�
x-1
�
= -x-2 = - 1
x2
• ddx
�
1
xc
�
= ddx (x
-c) = - c
xc+1
• ddx
p
x = ddxx
1
2 = 12x
- 1
2 = 1
2
p
x
,
Func¸o˜es Exponenciais e Logar´ıtmicas
• ddxex = ex
• ddx ln(x) = 1x
• ddxax = ax ln(a)
Func¸o˜es Trigonome´tricas
• ddx sen x = cos x
• ddx cos x = -sen x,
• ddx tg x = sec2 x
• ddx sec x = tg x sec x
• ddx cotg x = -cossec 2x
• ddx cossec x = -cossec x cotg x
Func¸o˜es Trigonome´tricas Inversas
• ddx arcsen x = 1p1-x2
• ddx arccos x = -1p1-x2
• ddx arctg x = 11+x2
• ddx arcsec x = 1|x|px2-1
• ddx arccotg x = -11+x2
• ddx arccossec x = -1|x|px2-1
Func¸o˜es Hiperbo´licas
• ddx senh x = cosh x = e
x+e-x
2
• ddx cosh x = senh x = e
x-e-x
2
• ddx tgh x = sech2 x
• ddx sech x = - tgh x sech x
• ddx cotgh x = - cossech2 x
Func¸o˜es Hiperbo´licas Inversas
• ddx csch x = - coth x cossech x
• ddx arcsenh x = 1px2+1
• ddx arccosh x = 1px2-1
• ddx arctgh x = 11-x2
• ddx arcsech x = -1xp1-x2
• ddx arccoth x = 11-x2
• ddx arccossech x = -1|x|p1+x2
1
Integrais
Regras de Integrac¸a˜o
• R cf(x)dx = c R f(x)dx
• R[f(x) + g(x)]dx = R f(x)dx+ R g(x)dx
• R f 0(x)g(x)dx = f(x)g(x)- R f(x)g 0(x)dx
Func¸o˜es Racionais
• R xn dx = xn+1n+1 + c para n 6= -1
•
Z
1
x
dx = ln |x| + c
•
Z
du
1+ u2
= arctgu+ c
•
Z
1
a2 + x2
dx =
1
a
arctg(x/a) + c
•
Z
du
1- u2
=
�
arctgh u+ c, se |u| < 1
arccotgh u+ c, se |u| > 1
=
1
2 ln
�� 1+u
1-u
��+ c
Func¸o˜es Logar´ıtmicas
• R ln xdx = x ln x- x+ c
• R loga x dx = x loga x- xlna + c
Func¸o˜es Irracionais
•
Z
dup
1- u2
= arcsenu+ c
•
Z
du
u
p
u2 - 1
= arcsec u+ c
•
Z
dup
1+ u2
= arcsenh u+ c
= ln |u+
p
u2 + 1| + c
•
Z
dup
1- u2
= arccosh u+ c
= ln |u+
p
u2 - 1| + c
•
Z
du
u
p
1- u2
= -arcsech |u| + c
•
Z
du
u
p
1+ u2
= -arccosech |u| + c
•
Z
1p
a2 - x2
dx = arcsen
x
a
+ c
•
Z
-1p
a2 - x2
dx = arccos
x
a
+ c
Func¸o˜es Trigonome´tricas
• R cos xdx = sen x+ c
• R sen xdx = - cos x+ c
• R tg xdx = ln |sec x| + c
• R csc xdx = ln |csc x- cot x| + c
• R sec xdx = ln |sec x+ tg x| + c
• R cot xdx = ln |sen x| + c
• R sec x tg xdx = sec x+ c
• R csc x cot xdx = - csc x+ c
• R sec2 x dx = tg x+ c
• R csc2 x dx = - cot x+ c
• R sen2 x dx = 12(x- sen x cos x) + c
• R cos2 x dx = 12(x+ sen x cos x) + c
Func¸o˜es Hiperbo´licas
• R sinh xdx = cosh x+ c
• R cosh xdx = sinh x+ c
• R tgh xdx = ln(cosh x) + c
• R csch xdx = ln ��tgh x2 ��+ c
• R sech xdx = arctg(sinh x) + c
• R coth xdx = ln | sinh x| + c
2
Identidades Trigonome´tricas
1. sen(90o - ✓) = cos ✓
2. cos(90o - ✓) = sen ✓
3.
sen ✓
cos ✓
= tg ✓
4. sen2 ✓+ cos2 ✓ = 1
5. sec2 ✓- tg2 ✓ = 1
6. csc2 ✓- cot2 ✓ = 1
7. sen 2✓ = 2 sen ✓ cos ✓
8. cos 2✓ = cos2 ✓- sen2 ✓ = 2 cos2 ✓- 1
9. sen 2✓ = 2 sen ✓ cos ✓
10. sen(↵± �) = sen↵ cos�± cos↵ sen�
11. cos(↵± �) = cos↵ sen�± sen↵ cos�
12. tg(↵± �) = tg↵± tg�
1⌥ tg↵ tg�
13. sen↵± sen� = 2 sen 1
2
(↵± �) cos 1
2
(↵± �)
14. cos↵+ cos� = 2 cos
1
2
(↵+ �) cos
1
2
(↵- �)
15. cos↵- cos� = 2 sen
1
2
(↵+ �) sen
1
2
(↵- �)
3