Formulario Calculo Infinitesimal

1
FORMULARIO DE CÁLCULO
DIFERENCIAL & INTEGRAL
Constantes: a, b, C, m, n
Funciones de "x": U, V, W
CÁLCULO DIFERENCIAL
001
d
dx
(a) = 0 002
d
dx
(xn) = n ⋅ xn−1
003
d
dx
(UN) = N ⋅ Un−1
⋅
dU
dx
004
d
dx
(U ⋅ V) = U ⋅
dV
dx
+
dU
dx
⋅ V 005 d
dx
(
U
V
) =
V ⋅
dU
dx
− U ⋅
dV
dx
V2
006
d
dx
(Ln[U]) =
1
U
⋅
dU
dx
007
d
dx
(Loga[U]) =
1
Ln(a)
⋅
1
U
⋅
dU
dx
008
d
dx
(℮U) = ℮U
⋅
dU
dx
009
d
dx
(aU) = Ln(a) ⋅ aU
⋅
dU
dx
010
d
dx
(UV) = V ⋅ UV−1
+ Ln(U) ⋅ UV
⋅
dV
dx
011
d
dx
(Sen[U]) = Cos[U] ⋅
dU
dx
012
d
dx
(Cos[U]) = −Sen[U] ⋅
dU
dx
013
d
dx
(Tan[U]) = Sec2[U] ⋅
dU
dx
014
d
dx
(Ctg[U]) = −Csc2[U] ⋅
dU
dx
015
d
dx
(Sec[U]) = Sec[U] ⋅ Tan[U] ⋅
dU
dx
016
d
dx
(Csc[U]) = −Csc[U] ⋅ Ctg[U] ⋅
dU
dx
017
d
dx
(∢Sen[U]) =
1
√1 − U2
⋅
dU
dx
018
d
dx
(∢Cos[U]) =
−1
√1 − U2
⋅
dU
dx
019
d
dx
(∢Tan[U]) =
1
1 + U2
⋅
dU
dx
020
d
dx
(∢Ctg[U]) =
−1
1 + U2
⋅
dU
dx
021
d
dx
(∢Sec[U]) =
1
U ⋅ √U2 − 1
⋅
dU
dx

2
022
d
dx
(∢Csc[U]) =
−1
U ⋅ √U2 − 1
⋅
dU
dx
TRIGONOMETRÍA PLANA
A1 Sen(θ) =
co
h
=
1
Csc(θ)
h2
= co2
+ ca2
θ + θc = 90°
A2 Cos(θ) =
ca
h
=
1
Sec(θ)
A3 Tan(θ) =
co
ca
=
1
Ctg(θ)
=
Sen(θ)
Cos(θ)
A4 Ctg(θ) =
ca
co
=
1
Tan(θ)
=
Cos(θ)
Sen(θ)
A5 Sec(θ) =
h
ca
=
1
Cos(θ)
A6 Csc(θ) =
h
co
=
1
Sen(θ)
B1
Sen2
(θ) + Cos2
(θ) = 1
Tan2
(θ) + 1 = Sec2
(θ)
1 + Cot2
(θ) = Csc2
(θ)
θ = ω ⋅ t
{
x = Cos[ω ⋅ t]
y = Sen[ω ⋅ t]
x2
+ y2
= 1
B2
Sen(−θ) = −Sen(θ)
Cos(−θ) = Cos(θ)
Tan(−θ) = −Tan(θ)
C1 Sen(α + β) = Sen(α) ⋅ Cos(β) + Cos(α) ⋅ Sen(β)
C2 Sen(α − β) = Sen(α) ⋅ Cos(β) − Cos(α) ⋅ Sen(β)
C3 Cos(α + β) = Cos(α) ⋅ Cos(β) − Sen(α) ⋅ Sen(β)
C4 Cos(α − β) = Cos(α) ⋅ Cos(β) + Sen(α) ⋅ Sen(β)
C5 Tan(α + β) =
Tan(α) + Tan(β)
1 − Tan(α) ⋅ Tan(β)
C6 Tan(α − β) =
Tan(α) − Tan(β)
1 + Tan(α) ⋅ Tan(β)
D1 Sen(2 ⋅ α) = 2 ⋅ Sen(α) ⋅ Cos(α)
D2 Cos(2 ⋅ α) = Cos2
(α) − Sen2
(α)

3
D3 Tan(2 ⋅ α) =
2 ⋅ Tan(α)
1 − Tan2(α)
E1 Sen2
(α) =
1
2
−
Cos(2 ⋅ α)
2
E2 Cos2
(α) =
1
2
+
Cos(2 ⋅ α)
2
F1 Sen(α) ⋅ Cos(β) =
Sen(α + β)
2
+
Sen(α − β)
2
F2 Cos(α) ⋅ Sen(β) =
Sen(α + β)
2
−
Sen(α − β)
2
F3 Cos(α) ⋅ Cos(β) =
Cos(α + β)
2
+
Cos(α − β)
2
F4 Sen(α) ⋅ Sen(β) = −
Cos(α + β)
2
+
Cos(α − β)
2
G1 Sen(α) + Sen(β) = 2 ⋅ Sen (
α + β
2
) ⋅ Cos (
α − β
2
)
G2 Sen(α) − Sen(β) = 2 ⋅ Cos (
α + β
2
) ⋅ Sen (
α − β
2
)
G3 Cos(α) + Cos(β) = 2 ⋅ Cos (
α + β
2
) ⋅ Cos (
α − β
2
)
G4 Cos(α) − Cos(β) = −2 ⋅ Sen (
α + β
2
) ⋅ Sen (
α − β
2
)
π = lim
N→∞
N ⋅ Tan (
180°
N
) = 3.141592 …
H1 Sen(θ) = θ −
θ3
3!
+
θ5
5!
−
θ7
7!
+ ⋯
H2 Cos(x) = 1 −
θ2
2!
+
θ4
4!
−
θ6
6!
+ ⋯
LOGARITMOS
Ln(x) ≡ Log℮(x)
Si: y = ax
⇒ x = Loga(y) =
Ln(y)
Ln(a)
Si: y = ℮x
⇒ x = Ln(y)
J1 Loga(U ⋅ V) = Loga(U) + Loga(V)
J2 Loga (
U
V
) = Loga(U) − Loga(V)

4
J3 Loga(Un) = n ⋅ Loga(U)
J4 Loga( √U
m
) =
1
m
⋅ Loga(U)
J5 Loga (U
n
m) =
n
m
⋅ Loga(U)
℮ = lim
N→∞
(1 +
1
N
)
N
= 2.718281 …
J6 ℮x
= 1 + x +
x2
2!
+
x3
3!
+
x4
4!
+
x5
5!
+
x6
6!
+
x7
7!
+ ⋯
CÁLCULO INTEGRAL
023 ∫ dx = x + C 024
∫ xn
⋅ dx =
xn+1
n + 1
+ C
n ≠ −1
025 ∫
dx
x
= Ln|x| + C 026
∫ Un
⋅ dU =
Un+1
n + 1
+ C
n ≠ −1
027 ∫
dU
U
= Ln|U| + C 028 ∫ ℮U
⋅ dU = ℮U
+ C
029 ∫ aU
⋅ dU =
aU
Ln|a|
+ C 030 ∫ U ⋅ dV = U ⋅ V − ∫ V ⋅ dU
031 ∫ Ln(U) ⋅ dU = U ⋅ Ln|U| − U + C
032 ∫ Sen(U) ⋅ dU = −Cos(U) + C 033 ∫ Cos(U) ⋅ dU = Sen(U) + C
034 ∫ Tan(U) ⋅ dU = Ln|Sec(U)| + C
035 ∫ Ctg(U) ⋅ dU = Ln|Sen(U)| + C
036 ∫ Sec(U) ⋅ dU = Ln|Sec(U) + Tan(U)| + C
037 ∫ Csc(U) ⋅ dU = Ln|Csc(U) − Ctg(U)| + C
038 ∫ Sec2
(U) ⋅ dU = Tan(U) + C

5
039 ∫ Csc2
(U) ⋅ dU = −Ctg(U) + C
040 ∫ Sec(U) ⋅ Tan(U) ⋅ dU = Sec(U) + C
041 ∫ Csc(U) ⋅ Ctg(U) ⋅ dU = −Csc(U) + C
042
∫ Sen(m ⋅ x) ⋅ Cos(n ⋅ x) ⋅ dx
= −
Cos([m + n] ⋅ x)
2 ⋅ (m + n)
−
Cos([m − n] ⋅ x)
2 ⋅ (m − n)
+ C
043
∫ Sen(m ⋅ x) ⋅ Sen(n ⋅ x) ⋅ dx
= −
Sen([m + n] ⋅ x)
2 ⋅ (m + n)
+
Sen([m − n] ⋅ x)
2 ⋅ (m − n)
+ C
044
∫ Cos(m ⋅ x) ⋅ Cos(n ⋅ x) ⋅ dx
= +
Sen([m + n] ⋅ x)
2 ⋅ (m + n)
+
Sen([m − n] ⋅ x)
2 ⋅ (m − n)
+ C
045 ∫ ∢Sen(U) ⋅ dU = U ⋅ ∢Sen(U) + √1 − U2 + C
046 ∫ ∢Cos(U) ⋅ dU = U ⋅ ∢Cos(U) − √1 − U2 + C
047 ∫ ∢Tan(U) ⋅ dU = U ⋅ ∢Tan(U) −
1
2
⋅ Ln|U2
+ 1| + C
048 ∫ ∢Ctg(U) ⋅ dU = U ⋅ ∢Ctg(U) +
1
2
⋅ Ln|U2
+ 1| + C
049 ∫ ∢Sec(U) ⋅ dU = U ⋅ ∢Sec(U) − Ln |U + √U2 − 1| + C
050 ∫ ∢Csc(U) ⋅ dU = U ⋅ ∢Csc(U) + Ln |U + √U2 − 1| + C
Forma: U2
+ a2
051 ∫ √U2 + a2 ⋅ dU =
U
2
⋅ √U2 + a2 +
a2
2
⋅ Ln |U + √U2 + a2| + C

6
052
∫ U2
⋅ √U2 + a2 ⋅ dU =
U
8
⋅ (2 ⋅ U2
+ a2) ⋅ √U2 + a2
−
a4
8
⋅ Ln |U + √U2 + a2| + C
053 ∫
√U2 + a2
U
⋅ dU = √U2 + a2 − a ⋅ Ln |
a + √U2 + a2
U
| + C
054 ∫
√U2 + a2
U2
⋅ dU = −
√U2 + a2
U
+ Ln |U + √U2 + a2| + C
055 ∫
dU
U2 + a2
=
1
a
⋅ ∢Tan (
U
a
) + C
056 ∫
dU
√U2 + a2
= Ln |U + √U2 + a2| + C1 = invSenh (
U
a
) + C2
057 ∫
U2
⋅ dU
√U2 + a2
=
U
2
⋅ √U2 + a2 −
a2
2
⋅ Ln |U + √U2 + a2| + C
058 ∫
dU
U ⋅ √U2 + a2
= −
1
a
⋅ Ln |
a + √U2 + a2
U
| + C
059 ∫
dU
U2 ⋅ √U2 + a2
= −
√U2 + a2
a2 ⋅ U
+ C
060
∫(U2
+ a2)n
⋅ dU =
U ⋅ (U2
+ a2)n
2 ⋅ n + 1
+
2 ⋅ n ⋅ a2
2 ⋅ n + 1
⋅ ∫(U2
+ a2)n−1
⋅ dU
n ≠ −
1
2
061
∫
dU
(U2 + a2)n
=
1
a2
⋅
U
(2 ⋅ n − 2) ⋅ (U2 + a2)n−1
+
1
a2
⋅ (
2 ⋅ n − 3
2 ⋅ n − 2
) ⋅ ∫
dU
(U2 + a2)n−1
n ≠ 1
Forma: U2
− a2
062 ∫ √U2 − a2 ⋅ dU =
U
2
⋅ √U2 − a2 −
a2
2
⋅ Ln |U + √U2 − a2| + C

7
063
∫ U2
⋅ √U2 − a2 ⋅ dU =
U
8
⋅ (2 ⋅ U2
− a2) ⋅ √U2 − a2
−
a4
8
⋅ Ln |U + √U2 − a2| + C
064 ∫
√U2 − a2
U
⋅ dU = √U2 − a2 − a ⋅ ∢Cos (
a
U
) + C
065 ∫
√U2 − a2
U2
⋅ dU = −
√U2 − a2
U
+ Ln |U + √U2 − a2| + C
066 ∫
dU
U2 − a2
=
1
2 ⋅ a
⋅ Ln |
U − a
U + a
| + C = − (
1
a
) ⋅ invCtgh (
U
a
) + C
067 ∫
dU
√U2 − a2
= Ln |U + √U2 − a2| + C1 = invCosh (
U
a
) + C2
068 ∫
U2
⋅ dU
√U2 − a2
=
U
2
⋅ √U2 − a2 +
a2
2
⋅ Ln |U + √U2 − a2| + C
069 ∫
dU
U ⋅ √U2 − a2
=
1
a
⋅ ∢Sec (
U
a
) + C
070 ∫
dU
U2 ⋅ √U2 − a2
=
√U2 − a2
a2 ⋅ U
+ C
071
∫(U2
− a2)n
⋅ dU =
U ⋅ (U2
− a2)n
2 ⋅ n + 1
−
2 ⋅ n ⋅ a2
2 ⋅ n + 1
⋅ ∫(U2
− a2)n−1
⋅ dU
n ≠ −
1
2
072
∫
dU
(U2 − a2)n
= −
1
a2
⋅
U
(2 ⋅ n − 2) ⋅ (U2 − a2)n−1
−
1
a2
⋅ (
2 ⋅ n − 3
2 ⋅ n − 2
) ⋅ ∫
dU
(U2 − a2)n−1
n ≠ 1
Forma: a2
− U2
073 ∫ √a2 − U2 ⋅ dU =
U
2
⋅ √a2 − U2 +
a2
2
⋅ ∢Sen (
U
a
) + C

8
074
∫ U2
⋅ √a2 − U2 ⋅ dU =
U
8
⋅ (2 ⋅ U2
− a2) ⋅ √a2 − U2 +
a4
8
⋅ ∢Sen (
U
a
) + C
075 ∫
√a2 − U2
U
⋅ dU = √a2 − U2 + a ⋅ Ln|U| − a ⋅ Ln |a + √a2 − U2| + C
076 ∫
√a2 − U2
U2
⋅ dU = −
√a2 − U2
U
− ∢Sen (
U
a
) + C
077 ∫
dU
a2 − U2
=
1
2 ⋅ a
⋅ Ln |
a + U
a − U
| + C = (
1
a
) ⋅ invTanh (
U
a
) + C
078 ∫
dU
√a2 − U2
= ∢Sen (
U
a
) + C
079 ∫
U2
⋅ dU
√a2 − U2
= −
U
2
⋅ √a2 − U2 +
a2
2
⋅ ∢Sen (
U
a
) + C
080 ∫
dU
U ⋅ √a2 − U2
=
1
a
⋅ Ln|U| −
1
a
⋅ Ln |a + √a2 − U2| + C
081 ∫
dU
U2 ⋅ √a2 − U2
= −
√a2 − U2
a2 ⋅ U
+ C
082
∫(a2
− U2)n
⋅ dU =
U ⋅ (a2
− U2)n
2 ⋅ n + 1
+
2 ⋅ n ⋅ a2
2 ⋅ n + 1
⋅ ∫(a2
− U2)n−1
⋅ dU
n ≠ −
1
2
083
∫
dU
(a2 − U2)n
=
1
a2
⋅
U
(2 ⋅ n − 2) ⋅ (a2 − U2)n−1
+
1
a2
⋅ (
2 ⋅ n − 3
2 ⋅ n − 2
) ⋅ ∫
dU
(a2 − U2)n−1
n ≠ 1
Forma: a + b ⋅ U
084 ∫
U ⋅ dU
a + b ⋅ U
=
a
b2
+
U
b
−
a
b2
⋅ Ln|a + b ⋅ U| + C

9
085
∫
U2
⋅ dU
a + b ⋅ U
=
1
2 ⋅ b3
⋅ (a + b ⋅ U)2
−
2 ⋅ a
b3
⋅ (a + b ⋅ U) +
a2
b3
⋅ Ln|a + b ⋅ U| + C
086 ∫
dU
U ⋅ (a + b ⋅ U)
=
1
a
⋅ Ln |
1
a + b ⋅ U
| + C
087 ∫
U ⋅ dU
(a + b ⋅ U)2
=
a
b2 ⋅ (a + b ⋅ U)
+
1
b2
⋅ Ln|a + b ⋅ U| + C
088 ∫
U2
⋅ dU
(a + b ⋅ U)2
=
a
b3
+
U
b2
−
a2
b3 ⋅ (a + b ⋅ U)
−
2 ⋅ a
b3
⋅ Ln|a + b ⋅ U| + C
089 ∫
U ⋅ dU
√a + b ⋅ U
=
2
3 ⋅ b2
⋅ (b ⋅ U − 2 ⋅ a) ⋅ √a + b ⋅ U + C
090 ∫
dU
U ⋅ √a + b ⋅ U
=
{
1
√a
⋅ Ln |
√a + b ⋅ U − √a
√a + b ⋅ U + √a
| + C, a > 0
2
√−a
⋅ ∢Tan√
a + b ⋅ U
−a
+ C, a < 0
091 ∫
dU
U2 ⋅ (a + b ⋅ U)
= −
1
a ⋅ U
+
b
a2
⋅ Ln |
a + b ⋅ U
U
| + C
092 ∫
dU
U ⋅ (a + b ⋅ U)2
=
1
a ⋅ (a + b ⋅ U)
−
1
a2
⋅ Ln |
a + b ⋅ U
U
| + C
093 ∫ U ⋅ √a + b ⋅ U ⋅ dU =
2
15 ⋅ b2
⋅ (3 ⋅ b ⋅ U − 2 ⋅ a) ⋅ (a + b ⋅ U)
3
2 + C
094
∫
U2
⋅ dU
√a + b ⋅ U
⋅ du =
2
15 ⋅ b3
⋅ (8 ⋅ a2
+ 3 ⋅ b2
⋅ U2
− 4 ⋅ a ⋅ b ⋅ U) ⋅ √a + b ⋅ U + C

10
095
∫
√a + b ⋅ U
U
⋅ dU
=
{
2 ⋅ √a + b ⋅ U + √a ⋅ Ln |
√a + b ⋅ U − √a
√a + b ⋅ U + √a
| + C, a > 0
2 ⋅ √a + b ⋅ U + 2 ⋅ √−a ⋅ ∢Tan√
a + b ⋅ U
−a
+ C, a < 0
096
∫
√a + b ⋅ U
U2
⋅ dU
=
{
−
√a + b ⋅ U
U
+
b
2 ⋅ √a
⋅ Ln |
√a + b ⋅ U − √a
√a + b ⋅ U + √a
| + C, a > 0
−
√a + b ⋅ U
U
+
b
√a
⋅ ∢Tan√
a + b ⋅ U
−a
+ C, a < 0
FÓRMULAS DE REDUCCIÓN
097
∫
dx
(a ⋅ x2 + b ⋅ x + c)n
=
2 ⋅ a ⋅ x + b
(n − 1) ⋅ (4 ⋅ a ⋅ c − b2) ⋅ (a ⋅ x2 + b ⋅ x + c)n−1
+
2 ⋅ a ⋅ (2 ⋅ n − 3)
(n − 1) ⋅ (4 ⋅ a ⋅ c − b2)
⋅ ∫
dx
(a ⋅ x2 + b ⋅ x + c)n−1
098 ∫ Un
⋅ ℮a⋅U
⋅ dU =
1
a
⋅ Un
⋅ ℮a⋅U
−
n
a
⋅ ∫ Un−1
⋅ ℮a⋅U
⋅ dU
TRIGONOMETRÍA HIPERBÓLICA
K1 Senh(U) =
℮U
− ℮−U
2
K2 Cosh(U) =
℮U
+ ℮−U
2
K3 Tanh(U) =
℮U
− ℮−U
℮U + ℮−U
=
Senh(U)
Cosh(U)
K4 Ctgh(U) =
℮U
+ ℮−U
℮U − ℮−U
=
Cosh(U)
Senh(U)
=
1
Tanh(U)
U ≠ 0

11
K5 Sech(U) =
2
℮U + ℮−U
=
1
Cosh(U)
K6 Csch(U) =
2
℮U − ℮−U
=
1
Senh(U)
U ≠ 0
L1
Cosh2
(θ) − Senh2
(θ) = 1
1 − Tanh2
(θ) = Sech2
(θ)
Coth2
(θ) − 1 = Csch2
(θ)
{
x = Cosh[t]
y = Senh[t]
x2
− y2
= 1
L2
Senh(−θ) = −Senh(θ)
Cosh(−θ) = Cosh(θ)
Tanh(−θ) = −Tanh(θ)
Si: y = Senh(x) =
℮x
− ℮−x
2
⇒ x = invSenh(y) = Ln (y + √y2 + 1)
M1 invSenh(U) = Ln (U + √U2 + 1)
M2 invCosh(U) = Ln (U + √U2 − 1) U ≥ 1
M3 invTanh(U) =
1
2
⋅ Ln (
1 + U
1 − U
) U2
< 1
M4 invCtgh(U) =
1
2
⋅ Ln (
U + 1
U − 1
) U2
> 1
M5 invSech(U) = Ln (
1 + √1 − U2
U
) 0 < U ≤ 1
M6 invCsch(U) = Ln (
1
U
+
√U2 + 1
|U|
) U ≠ 0
099
d
dx
(Senh[U]) = Cosh[U] ⋅
dU
dx
100
d
dx
(Cosh[U]) = Senh[U] ⋅
dU
dx
101
d
dx
(Tanh[U]) = Sech2
[U] ⋅
dU
dx

12
102
d
dx
(Ctgh[U]) = −Csch2
[U] ⋅
dU
dx
103
d
dx
(Sech[U]) = −Sech[U] ⋅ Tanh[U] ⋅
dU
dx
104
d
dx
(Csch[U]) = −Csch[U] ⋅ Ctgh[U] ⋅
dU
dx
105
d
dx
(invSenh[U]) =
1
√U2 + 1
⋅
dU
dx
106
d
dx
(invCosh[U]) =
1
√U2 − 1
⋅
dU
dx
U > 1
107
d
dx
(invTanh[U]) =
1
1 − U2
⋅
dU
dx
U2
< 1
108
d
dx
(invCtgh[U]) =
1
1 − U2
⋅
dU
dx
U2
> 1
109
d
dx
(invSech[U]) =
−1
U ⋅ √1 − U2
⋅
dU
dx
0 < U < 1
110
d
dx
(invCsch[U]) =
−1
|U| ⋅ √U2 + 1
⋅
dU
dx
U ≠ 0
111 ∫ Senh(U) ⋅ dU = Cosh(U) + C
112 ∫ Cosh(U) ⋅ dU = Senh(U) + C
113 ∫ Tanh(U) ⋅ dU = Ln|Cosh(U)| + C
114 ∫ Ctgh(U) ⋅ dU = Ln|Senh(U)| + C
115 ∫ Sech(U) ⋅ dU = ∢Tan(Senh[U]) + C
116 ∫ Csch(U) ⋅ dU = Ln |Tanh (
U
2
)| + C
117 ∫ Sech2
(U) ⋅ dU = Tanh(U) + C
118 ∫ Csch2
(U) ⋅ dU = −Ctgh(U) + C

13
119 ∫ Sech(U) ⋅ Tanh(U) ⋅ dU = −Sech(U) + C
120 ∫ Csch(U) ⋅ Ctgh(U) ⋅ dU = −Csch(U) + C
Elaboró: MCI José A. Guasco.
https://www.slideshare.net/AntonioGuasco1/

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