Tabela de derivadas e integrais

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Resumo das fo´rmulas do Ca´lculo Prof. Ph.D. Marcelo Trindade
Derivadas elementares
1. ddxc = 0
2. ddxx
n = nxn−1
3. ddxa
x = ax(lnx)
4. ddx loga x =
1
x loga e
Derivadas trig.
1. ddx sin (x) = cos (x)
2. ddx cos (x) = − sin (x)
3. ddx tan (x) = sec
2 (x)
4. ddx cot (x) = − csc2 (x)
5. ddx sec (x) = sec (x) tan (x)
6. ddx csc (x) = − csc (x) cot (x)
7. ddx arcsin (x) =
1√
1−x2
8. ddx arccos (x) =
−1√
1−x2
9. ddx arctan (x) =
1√
1+x2
10. ddxarccot(x) =
−1√
1+x2
11. ddxarcsec(x) =
1
|x|√x2−1
12. ddxarccsc(x) =
−1
|x|√x2−1
Derivadas hiperbo´licas
1. ddx sinhx = cosh(x) =
ex+e−x
2
2. ddx cosh(x) = sinh(x) =
ex−e−x
2
3. ddx tanh(x) = sech
2(x)
4. ddx sech(x) = − tanh(x)sech(x)
5. ddxcotgh(x) = −csch2(x)
Regras de derivac¸a˜o
1. ddxfg = f
′g + fg′
2. ddx
f
g =
f ′g−g′f
g2
3. ddxf
g = gfg−1f ′ + fg(ln f)g′
4. ddxf(g) = f
′(g)g′
Integrais elementares
1.
∫
dx = x+ c
2.
∫
xndx = x
n+1
n+1 + c, n 6= −1
3.
∫
1
xdx = ln |x|+ c
4.
∫
axdx = a
x
ln a + c, a > 0, a 6= 1
5.
∫
lnxdx = x lnx− x+ c
6.
∫
exdx = ex + c
7.
∫
dx
x2+a2 =
1
a arctan
x
a + c
8.
∫
dx
x2−a2 =
1
2a ln |x−ax+a |+ c, x2 > a2
9.
∫
dx√
x2+a2
= ln |x+√x2 + a2|+ c
10.
∫
dx√
x2−a2 = ln |x+
√
x2 − a2|+ c
11.
∫
dx√
a2−x2 = arcsin
x
a + c, x
2 < a2
12.
∫
dx
x
√
a2−x2 =
1
a arcsec |xa |+ c
Integrais trig.
1.
∫
sin (x)dx = − cos (x) + c
2.
∫
cos (x)dx = sin (x) + c
3.
∫
tan (x)dx = ln | sec (x)|+ c
4.
∫
cot (x)dx = ln | sin (x)|+ c
5.
∫
sec (x)dx = ln | sec (x) + tan (x)|+c
6.
∫
csc (x)dx = ln | csc (x)− cot (x)|+c
7.
∫
sec (x) tan (x)dx = sec (x) + c
8.
∫
csc (x) cot (x)dx = − csc (x) + c
9.
∫
sec2(x)dx = tan (x) + c
10.
∫
csc2 (x)dx = − cot (x) + c
11.
∫
sin2(x)dx = 12 (x−sin(x) cos(x))+c
12.
∫
cos2(x)dx = 12 (x+sin(x) cos(x))+c
13.
∫
tan2(x)dx = tan(x)− x+ c
Integrais hiperbo´licas
1.
∫
sinh(x)dx = cosh(x) + c
2.
∫
cosh(x)dx = sinh(x) + c
3.
∫
tanh(x)dx = ln | cosh(x)|+ c
4.
∫
csch(x)dx = ln | tanh(x/2)|+ c
5.
∫
sech(x)dx = arctan(sinh(x)) + c
6.
∫
coth(x)dx = ln | sinh(x)|+ c
Regras de integrac¸a˜o
1.
∫
f(g)g′dx = F (g) + c
2.
∫
fgdx = fG− ∫ Gf ′
Recorreˆncias
1.
∫
sinn(ax)dx = − sinn−1(ax) cos (ax)an +
(n−1n )
∫
sinn−2(ax)dx
2.
∫
cosn(ax)dx = sin (ax) cos
n−1(ax)
an +
n−1
b
∫
cosn−2(ax)dx
3.
∫
tann(ax)dx = tan
n−1 (ax)
a(n−1) −∫
tann−2(ax)dx
4.
∫
cotn(ax)dx = − cotn−1 (ax)a(n−1) −∫
cotn−2(ax)dx
5.
∫
secn(ax)dx = sec
n−2 (ax) tan (ax)
a(n−1) +
n−2
n−1
∫
secn−2(ax)dx
6.
∫
cscn(ax)dx = − cscn−2 (ax) cot (ax)a(n−1) +
n−2
n−1
∫
cscn−2(ax)dx
Identidades
1. sin2(x) + cos2(x) = 1
2. 1 + tan2(x) = sec2(x)
3. 1 + cot2(x) = csc2(x)
4. sin2(x) = 1−cos(2x)2
5. cos2(x) = 1+cos(2x)2
6. sin(2x) = 2 sin (x) cos(x)
7. 2 sin(x) cos(y) = sin(x−y)+sin(x+y)
8. 2 sin(x) sin(y) = cos(x−y)−cos(x+y)
9. cos(x) cos(y) = cos(x−y)+cos(x+y)
10. 1± sin(x) = 1± cos(pi2 − x)
Substituic¸o˜es
1.
√
x2 + a2 = a sec(x)→ x = a tan(x)
2.
√
x2 − a2 = a tan(x)→ x = a sec(x)
3.
√
a2 − x2 = a cos(x)→ x = a sin(x)
Universidade Federal do Pampa
Resumo das fo´rmulas do Ca´lculo Prof. Ph.D. Marcelo Trindade
Integrais definidas
1.
∫ b
a
fdx = F (b)− F (a)
2.
∫ +∞
a
fdx = lim
b→+∞
∫ b
a
fdx
3.
∫ b
−∞ fdx = lima→−∞
∫ b
a
fdx
4.
∫ +∞
−∞ fdx =
lim
a→−∞
∫ c
a
fdx+ lim
b→+∞
∫ b
c
fdx
Aplicac¸o˜es das integrais
1. a´rea em coordenadas retangulares
A =
∫ b
a
f(x)dx
A =
∫ b
a
f(x)− g(x)dx
2. comprimento de curva
L =
∫ b
a
√
1 + f ′(x)2dx
L =
∫ β
α
√
f(θ)2 + f ′(θ)2dθ
3. a´rea em coordenadas polares
A = 12
∫ β
α
f(θ)2dθ
A = 12
∫ β
α
f(θ)2 − g(θ)2dθ
4. volume de so´lidos de revoluc¸a˜o
V = pi
∫ b
a
f(x)2dx
V = pi
∫ b
a
f(x)2 − g(x)2dx
5. a´rea de superf´ıcie de revoluc¸a˜o
A = 2pi
∫ b
a
f(x)
√
1 + f ′(x)2dx
Func¸o˜es vetoriais
1. div(~h) = ∂hi∂x +
∂hj
∂y +
∂hk
∂z
2. rot(~h) =
(
∂hk
∂y − ∂hj∂z
)
~i +(
∂hi
∂z − ∂hk∂x
)
~j +
(
∂hj
∂x − ∂hi∂y
)
~k
3. S =
∫∫
R
∣∣∣∣∣∣∣∣ ∂s∂u × ∂s∂v ∣∣∣∣∣∣∣∣dA
4. a(x − xo) + b(y − yo) + c(z − z0) =
0, < a~i, b~j, c~k >= ∂s∂u × ∂s∂v
5.
∫
C
~f • dr = ∫ b
a
~f(r(t)) • r′(t)dt
6. ∇φ = ∂φ∂x~i + ∂φ∂y~j + ∂φ∂z ~k
7.
∮
C
~f • dr = ∮
C
fi(x, y)dx +
fj(x, y)dy =
∫∫
R
(
∂fj
∂x − ∂fi∂y
)
dA
Universidade Federal do Pampa
	blue Derivadas elementares
	blue Derivadas trig.
	blueDerivadas hiperbólicas
	blue Regras de derivação
	blue Integrais elementares
	blue Integrais trig.
	blueIntegrais hiperbólicas
	blue Regras de integração
	blue Recorrências
	blue Identidades
	blue Substituições
	blue Integrais definidas
	blue Aplicações das integrais
	blue Funções vetoriais