Regras para Derivadas e Integrais

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Regras para derivadas
As funções y, u, v, w são funções da variavel x e as letras k, a, n são consideradas constantes.
A letra L representa o logaritmo neperiano.
(1)y = k → y′ = 0 (2)y = x → y′ = 1
→
(3)y = xn → y′ = n.xn−1 (4)y = k.xn → y′ = k.n.xn−1
→
(5)y = un → y′ = n.un−1.u′ (6)y = u + v → y′ = u′ + v′
→
(7)y = a.u → y′ = a.u′ (8)y = u.v → y′ = u′.v + u.v′
→
(9)y = au → y′ = au.L(a).u′ (10)y = eu → y′ = eu.u′
→
(11)y = logau → y′ =
u′
u.logea
(12)y = Lu → y′ = u
′
u
→
(13)y = uv → y′ = v.uv−1.u′ + uv.v′.L(u) (14)y = u
v
→ y′ = vu
′ − uv′
v2
→
(15)y = f(u);u = f(v)→ dy
dv
=
dy
du
.
du
dv
(16)x = f1(t), y = f2(t)→
dy
dx
=
dy/dt
dx/dt
→
(17)y = sen(u) → y′ = u′.cos(u) (18)y = cos(u) → y′ = −u′.sen(u)
→
(19)y = tg(u) → y′ = u′.sec2(u) (20)y = cotg(u) → y′ = −u′.cossec2(u)
→
(21)y = sec(u) → y′ = u′.sec(u)tg(u) (22)y = cossec(u) → y′ = −u′.cossec(u)cotg(u)
→
(23)y = arc sen(u) → y′ = u
′
√
1− u2
(24)y = arc cos(u) → y′ = −u
′
√
1− u2
→
(25)y = arc tg(u) → y′ = u
′
1 + u2
(26)y = arc cotg(u) → y′ = −u
′
1 + u2
→
(27)y = arc sec(u) → y′ = u
′
u(u2 − 1)
(28)f ′(x) = g′(x) → f(x) = g(x) + k
1
Regras para Integrais
As funções y, u, v, w são funções da variavel x e as letras k, a, n, c são consideradas constantes.
(1)
∫
dx = x + c (2)
∫
xndx =
xn+1
n + 1
+ c, n 6= −1
(3)
∫
sen kx dx =
−cos kx
k
+ c (4)
∫
cos kx dx =
sen kx
k
+ c
(5)
∫
sec2 x dx = tg k + c (6)
∫
cossec2 x dx = −cotg x + c
(7)
∫
sec x tg x dx = sec x + c (8)
∫
cossec xcotg x dx = −cossec x + c
(9)
∫
ekxdx =
ekx
k
+ c (10)
∫ 1
x
dx = ln|x|+ c
(11)
∫ dx√
1− x2
= arc sen x + c (12)
∫ dx
1− x2
= arc tg x + c
(13)
∫ dx
x(
√
x2 − 1)
= arc sec x + c (14)
∫
axdx = (
1
ln a
)ax, a > 0, a 6= 1
(15)
∫
undu =
un+1
n + 1
+ c, n 6= −1 (16)
∫ 1
u
du = ln|u|+ c
(17)
∫
tg u du = −ln|cos u|+ c (18)
∫
cotg u du = ln|sen u|+ c
(19)
∫
eudu = eu + c (20)
∫
audu =
au
ln|a|
+ c
(21)
∫
sen(u)du = −cos(u) + c (22)
∫
cos(u)du = sen(u) + c
(23)
∫
tg(u)du = ln|sec(u)|+ c (24)
∫
cotg(u)du = ln|sen(u)|+ c
(25)
∫
sec(u)tg(u)du = sec(u) + c (26)
∫
cossec(u)cotg(u)du = −cossec(u) + c
(27)
∫
sec(u)du = ln|sec(u) + tg(u)|+ c (28)
∫
cossec(u)du = ln|cossec(u)− cotg(u)|+ c
(29)
∫
sec2(u)du = tg(u) + C (30)
∫
cossec2(u)du = cotg(u) + C
(31)
∫ du√
a2 − u2
= arc sen(
u
a
) + c (32)
∫ du
a2 + u2
=
1
a
arc tg(
u
a
) + c
(33)
∫ du√
u2 ± a2
= ln|u +
√
u2 ± a2|+ c (34)
∫ du
u2 − a2
=
1
2a
ln|u− a
u + a
|+ c
(35)
∫ du
u(
√
u2 − a2)
=
1
a
arc sec(
u
a
) (36)
∫
u dv = u.v −
∫
v.du
2