Tabela de Derivadas e Integrais

UNIVERSIDADE FEDERAL DO ABC
Tabela de Derivadas, Integrais e Identidades Trigonométricas
Derivadas
Funções Trigonométricas Inversas
Regras de Derivação
• (cf(x)) 0 = cf 0 (x)
•
d
dx
arcsen x =
•
d
dx
arccos x =
•
d
dx
arctg x =
•
d
dx
arcsec x =
•
d
dx
arccotg x =
•
d
dx
arccossec x =
√ 1
1−x2
• Derivada da Soma
(f(x) + g(x)) 0 = f 0 (x) + g 0 (x)
• Derivada do Produto
0
0
0
(f(x)g(x)) = f (x)g(x) + f(x)g (x)
• Derivada do Quociente
f(x) 0 f 0 (x)g(x) − f(x)g 0 (x)
=
g(x)
g(x)2
• Regra da Cadeia
(f(g(x)) 0 = (f 0 (g(x))g 0 (x)
Funções Simples
•
d
dx c
=0
•
d
dx x
=1
•
d
dx cx
=c
d c
dx x
Funções Exponenciais e Logarı́tmicas
•
d x
dx e
•
d
dx
•
d x
dx a
√1
|x| x2 −1
−1
1+x2
√−1
|x| x2 −1
•
d
dx
senh x = cosh x =
ex +e−x
2
•
d
dx
cosh x = senh x =
ex −e−x
2
•
d
dx
tgh x = sech2 x
•
d
dx
sech x = − tgh x sech x
•
d
dx
cotgh x = − cossech2 x
Funções Hiperbólicas Inversas
= ex
ln(x) =
1
1+x2
Funções Hiperbólicas
= cxc−1
d
d 1
−1 = −x−2 = − 1
• dx
x = dx x
x2
d
1
d
−c ) = − c
• dx
xc = dx (x
xc+1
√
1
d
d 12
1
• dx
x = dx
x = 21 x− 2 = 2√
,
x
•
√ −1
1−x2
1
x
= ax ln(a)
Funções Trigonométricas
•
d
dx
sen x = cos x
•
d
dx
cos x = −sen x,
•
d
dx
tg x = sec2 x
•
d
dx
sec x = tg x sec x
•
d
dx
cotg x = −cossec
•
d
dx
cossec x = −cossec x cotg x
2x
1
•
d
dx
csch x = − coth x cossech x
•
d
dx
arcsenh x =
√ 1
x2 +1
•
d
dx
arccosh x =
√ 1
x2 −1
•
d
dx
arctgh x =
•
d
dx
arcsech x =
√−1
x 1−x2
•
d
dx
arccoth x =
1
1−x2
•
d
dx
arccossech x =
1
1−x2
√−1
|x| 1+x2
Integrais
Z
Regras de Integração
•
R
cf(x) dx = c f(x) dx
R
R
R
• [f(x) + g(x)] dx = f(x) dx + g(x) dx
R
R
• f 0 (x)g(x) dx = f(x)g(x) − f(x)g 0 (x) dx
•
R
Z
•
Z
•
Funções Racionais
•
R
Z
•
Z
•
Z
xn dx =
xn+1
n+1
+c
•
1
dx = ln |x| + c
x
•
du
= arctg u + c
1 + u2
•
•
•
=
•
•
Funções Logarı́tmicas
•
•
R
R
•
ln x dx = x ln x − x + c
loga x dx = x loga x −
•
x
ln a
+c
•
•
Funções Irracionais
Z
•
Z
•
Z
du
√
= arcsen u + c
1 − u2
•
du
√
= arcsec u + c
u u2 − 1
−1
x
√
dx = arccos + c
2
2
a
a −x
R
R
R
R
R
R
R
R
R
R
R
R
cos x dx = sen x + c
sen x dx = − cos x + c
tg x dx = ln |sec x| + c
csc x dx = ln |csc x − cot x| + c
sec x dx = ln |sec x + tg x| + c
cot x dx = ln |sen x| + c
sec x tg x dx = sec x + c
csc x cot x dx = − csc x + c
sec2 x dx = tg x + c
csc2 x dx = − cot x + c
sen2 x dx = 12 (x − sen x cos x) + c
cos2 x dx = 12 (x + sen x cos x) + c
Funções Hiperbólicas
•
du
√
= arcsenh u + c
1 + u2√
= ln |u + u2 + 1| + c
Z
du
• √
= arccosh u + c
1 − u2√
= ln |u + u2 − 1| + c
Z
du
√
•
= −arcsech |u| + c
u 1 − u2
•
x
1
√
dx = arcsen + c
2
2
a
a −x
Funções Trigonométricas
para n 6= −1
1
1
•
dx = arctg(x/a) + c
2
2
a +x
a
Z
du
arctgh u + c, se |u| < 1
=
•
2
arccotgh u + c, se |u| > 1
1 − u 1
1+u 2 ln 1−u + c
du
√
= −arccosech |u| + c
u 1 + u2
•
•
R
R
R
sinh x dx = cosh x + c
cosh x dx = sinh x + c
tgh x dx = ln(cosh x) + c
R
• csch x dx = ln tgh x2 + c
R
• sech x dx = arctg(sinh x) + c
R
• coth x dx = ln | sinh x| + c
2
Identidades Trigonométricas
1. sen(90o − θ) = cos θ
9. sen 2θ = 2 sen θ cos θ
2. cos(90o − θ) = sen θ
3.
10. sen(α ± β) = sen α cos β ± cos α sen β
11. cos(α ± β) = cos α cos β ∓ sen α sen β
sen θ
= tg θ
cos θ
2
12. tg(α ± β) =
2
4. sen θ + cos θ = 1
tg α ± tg β
1 ∓ tg α tg β
1
1
13. sen α ± sen β = 2 sen (α ± β) cos (α ± β)
2
2
5. sec2 θ − tg2 θ = 1
6. csc2 θ − cot2 θ = 1
7. sen 2θ = 2 sen θ cos θ
1
1
14. cos α + cos β = 2 cos (α + β) cos (α − β)
2
2
8. cos 2θ = cos2 θ − sen2 θ = 2 cos2 θ − 1
1
1
15. cos α − cos β = 2 sen (α + β) sen (α − β)
2
2
3