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Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
Fórmulas de
Cálculo Diferencial
e Integral VER.6.8
Jesús Rubí Miranda (jesusrubim@yahoo.com)
http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO
1 1
1 1
si 0
si 0
y
0 y 0 0
ó
ó
n n
k k
k k
n n
k k
k k
a a
a
a a
a a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥⎧
= ⎨
− <⎩
= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
EXPONENTES
( )
( )
/
p q p q
p
p q
q
qp pq
p p p
p p
p
qp q p
a a a
a
a
a
a a
a b a b
a a
b b
a a
+
−
⋅ =
=
=
⋅ = ⋅
⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
LOGARITMOS
10
log
log log log
log log log
log log
log ln
log
log ln
log log y log ln
x
a
a a a
a a a
r
a a
b
a
b
e
N x a
MN M N
M
M N
N
N r N
N N
N
a a
N
N N N N
= ⇒
= +
= −
=
= =
= =
=
ALGUNOS PRODUCTOS
ad+( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
2 2
2 2 2
2 2 2
2
2
3 3 2 2 3
3 3 2 2 3
2 2 2 2
2
2
3 3
3 3
2 2 2
a c d ac
a b a b a b
a b a b a b a ab b
a b a b a b a ab b
x b x d x b d x bd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
a b a a b ab b
a b a a b ab b
a b c a b c ab ac bc
⋅ + =
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + + + +
1
1
n
n k k n n
k
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a b a b n− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
⎛ ⎞
− ⋅ = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
( ) ( )
( ) ( )
( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
5 4 3 2 2 3 4 5 6 6
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a a b a b a b ab b a b
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
( ) ( )
( ) ( )
1 1
1
1 1
1
1 impar
1 par
n
k n k k n n
k
n
k n k k n n
k
a b a b a b n
a b a b a b n
+ − −
=
+ − −
=
⎛ ⎞
+ ⋅ − = + ∀ ∈⎜ ⎟
⎝ ⎠
⎛ ⎞
+ ⋅ − = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
∑
SUMAS Y PRODUCTOS
n
( )
( )
1 2
1
1
1 1
1 1 1
1 0
n k
k
n
k
n n
k k
k k
n n n
k k k k
k k k
n
k k n
k
a a a a
c nc
ca c a
a b a b
a a a a
=
=
= =
= = =
−
=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
( )
1
( )
( )
( )
( )
( )
( )
( )
1
1
1
2
1
2 3 2
1
3 4 3 2
1
4 5 4 3
1
2
1
1 2 1
2
=
2
1
1 1
1
2
1
2 3
6
1
2
4
1
6 15 10
30
1 3 5 2 1
!
n
k
nn
k
k
n
k
n
k
n
k
n
k
n
k
n
a k d a n d
n
a l
r a rl
ar a
r r
k n n
k n n n
k n n n
k n n n n
n n
n k
n n
k
=
−
=
=
=
=
=
=
+ − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
+
− −
= =
− −
= +
= + +
= + +
= + + −
+ + + + − =
=
⎛ ⎞
=⎜ ⎟
⎝ ⎠
∑
∑
∑
∑
∑
∑
∏
( )
( )
0
!
,
! !
n
n n k k
k
k n
n k k
n
x y x y
k
−
=
≤
−
⎛ ⎞
+ = ⎜ ⎟
⎝ ⎠
∑
( ) 1 2
1 2 1 2
1 2
!
! ! !
k
n nn n
k k
k
n
x x x x x x
n n n
+ + + = ⋅∑
CONSTANTES
9…3.1415926535
2.71828182846e
π =
= …
TRIGONOMETRÍA
1
sen csc
sen
1
cos sec
cos
sen 1
tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θ θ
θ
θ θ
θ
θ
θ θ
θ θ
= =
= =
= = =
radianes=180π
CA
CO
HIP
θ
θ sin cos tg ctg sec csc
0 0 1 0 ∞ 1 ∞
30 1 2 3 2 1 3 3 2 3 2
45 1 2 1 2 1 1 2 2
60 3 2 1 2 3 1 3 2 2 3
90 1 0 ∞ 0 ∞ 1
[ ]
[ ]
sin ,
2 2
cos 0,
tg ,
2 2
1
ctg tg 0,
1
sec cos 0,
1
csc sen ,
2 2
y x y
y x y
y x y
y x y
x
y x y
x
y x y
x
π π
π
π π
π
π
π π
⎡ ⎤
= ∠ ∈ −⎢ ⎥
⎣ ⎦
= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
⎡ ⎤
= ∠ = ∠ ∈ −⎢ ⎥
⎣ ⎦
Gráfica 1. Las funciones trigonométricas: sin x ,
cos x , tg x :
-8 -6 -4 -2 0 2 4 6 8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sen x
cos x
tg x
Gráfica 2. Las funciones trigonométricas csc x ,
sec x , ctg x :
-8 -6 -4 -2 0 2 4 6 8
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
csc x
sec x
ctg x
Gráfica 3. Las funciones trigonométricas inversas
arcsin x , arccos x , arctg x :
-3 -2 -1 0 1 2 3
-2
-1
0
1
2
3
4
arc sen x
arc cos x
arc tg x
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x :
-5 0 5
-2
-1
0
1
2
3
4
arc ctg x
arc sec x
arc csc x
IDENTIDADES TRIGONOMÉTRICAS
2
2
2 2
2 2
sin cos 1
1 ctg csc
tg 1 sec
θ θ
θ θ
θ θ
+ =
+ =
+ =
( )
( )
( )
sin sin
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
sin 2 sin
cos 2 cos
tg 2 tg
sin sin
cos cos
tg tg
sin 1 sin
cos 1 cos
tg tg
n
n
n
n
n
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( )
( ) ( )
( )
( )
sin 0
cos 1
tg 0
2 1
sin 1
2
2 1
cos 0
2
2 1
tg
2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+⎛ ⎞
= −⎜ ⎟
⎝ ⎠
+⎛ ⎞
=⎜ ⎟
⎝ ⎠
+⎛ ⎞
= ∞⎜ ⎟
⎝ ⎠
sin cos
2
cos sin
2
π
θ θ
π
θ θ
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
( )
( )
( )
( )
( )
2 2
2
2
2
2
sin sin cos cos sin
cos cos cos sin sin
tg tg
tg
1 tg tg
sin 2 2sin cos
cos2 cos sin
2tg
tg2
1 tg
1
sin 1 cos2
2
1
cos 1 cos2
2
1 cos2
tg
1 cos2
α β α β α β
α β α β α β
α β
α β
α β
θ θ θ
θ θ θ
θ
θ
θ
θ θ
θ θ
θ
θ
θ
± = ±
± =
±
± =
=
= −
=
−
= −
= +
−
=
+
∓
∓
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1
sin sin 2sin cos
2 2
1 1
sin sin 2sin cos
2 2
1 1
cos cos 2cos cos
2 2
1 1
cos cos 2sin sin
2 2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
( )sin
tg tg
cos cos
α β
α β
α β
±
± =
⋅
( ) ( )
( ) ( )
( ) ( )
1
sin cos sin sin
2
1
sin sin cos cos
2
1
cos cos cos cos
2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
tg tg
tg tg
ctg ctg
α β
α β
α β
+
⋅ =
+
FUNCIONES HIPERBÓLICAS
sinh
2
cosh
2
sinh
tgh
cosh
1
ctgh
tgh
1 2
sech
cosh
1 2
csch
sinh
x x
x x
x
x
x x
x x
x x
x
e e
x
x e e
x
x e e
e e
x
x e e
x
x e e
x
x e e
−
−
−
−
−
−
−
=
+
=
−
= =
+
+
= =
−
= =
+
= =
−
x x
e e−
−
x
x
[
{ }
]
{ } { }
sinh :
cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
Gráfica 5. Las funciones hiperbólicas sinh x ,
cosh x , tgh x :
-5 0 5
-4
-3
-2
-1
0
1
2
3
4
5
senh x
cosh x
tgh x
FUNCIONES HIPERBÓLICAS INV
( )
( )
1 2
1 2
1
1
2
1
2
1
sinh ln 1 ,
cosh ln 1 , 1
1 1
tgh ln , 1
2 1
1 1
ctgh ln , 1
2 1
1 1
sech ln , 0 1
1 1
csch ln , 0
x x x x
x x x x
x
x x
x
x
x x
x
x
x x
x
x
x x
x x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞
= <⎜ ⎟
−⎝ ⎠
+⎛ ⎞
= >⎜ ⎟
−⎝ ⎠
⎛ ⎞± −
⎜ ⎟= < ≤
⎜ ⎟
⎝ ⎠
⎛ ⎞+
⎜ ⎟= + ≠
⎜ ⎟
⎝ ⎠
Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
IDENTIDADES DE FUNCS HIP
2 2
sinh 1x x
( )
( )
( )
2 2
2 2
cosh
1 tgh sech
ctgh 1 csch
sinh sinh
cosh cosh
tgh tgh
x x
x x
x x
x x
x x
− =
− =
− = −
− =
− = −
− =
( )
( )
( )
2 2
2
sinh sinh cosh cosh sinh
cosh cosh cosh sinh sinh
tgh tgh
tgh
1 tgh tgh
sinh 2 2sinh cosh
cosh 2 cosh sinh
2tgh
tgh 2
1 tgh
x y x y x y
x y x y x y
x y
x y
x y
x x x
x x x
x
x
x
± = ±
± = ±
±
± =
±
=
= +
=
+
( )
( )
2
2
2
1
sinh cosh 2 1
2
1
cosh cosh 2 1
2
cosh 2 1
tgh
cosh 2 1
x x
x x
x
x
x
= −
= +
−
=
+
sinh 2
tgh
cosh 2 1
x
x
x
=
+
cosh sinh
cosh sinh
x
x
e x x
e x x−
= +
= −
OTRAS
( ) ( )
2
2
2
0
4
2
4 discriminante
exp cos sin si ,
ax bx c
b b ac
x
a
b ac
i e iα
α β β β α β
+ + =
− ± −
⇒ =
− =
± = ± ∈
LÍMITES
( )
1
0
0
0
0
1
lim 1 2.71828...
1
lim 1
sen
lim 1
1 cos
lim 0
1
lim 1
1
lim 1
ln
x
x
x
x
x
x
x
x
x
x e
e
x
x
x
x
x
e
x
x
x
→
→∞
→
→
→
→
+ = =
⎛ ⎞
+ =⎜ ⎟
⎝ ⎠
=
−
=
−
=
−
=
DERIVADAS
( )
( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x
x x
n n
f x x f xdf y
D f x
dx x x
d
c
dx
d
cx c
dx
d
cx ncx
dx
d du dv dw
u v w
dx dx dx dx
d du
cu c
dx dx
∆ → ∆ →
−
+ ∆ − ∆
= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
( )
( )
( ) ( )
( )
2
1n n
d dv du
uv u v
dx dx dx
d dw dv du
uvw uv uw vw
dx dx dx dx
v du dx u dv dxd u
dx v v
d du
u nu
dx dx
−
= +
= + +
−⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
( )
( )
( )
( )
12
1 2
(Regla de la Cadena)
1
donde
dF dF du
dx du dx
du
dx dx du
dF dudF
dx dx du
x f tf tdy dtdy
dx dx dt f t y f t
= ⋅
=
=
=⎧′ ⎪
= = ⎨
′ =⎪⎩
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( ) 1
ln
log
log
log
log 0, 1
ln
ln
a
a
u u
u u
v v v
u
dx u u dx
d e du
u
dx u dx
ed du
u a
dx u dx
d du
e e
dx dx
d du
a a a
dx dx
d du dv
u vu u u
dx dx dx
−
= = ⋅
= ⋅
= ⋅ >
= ⋅
= ⋅
= + ⋅ ⋅
1d du dx du
a ≠
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
sin cos
cos sin
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
d du
u u
dx dx
=
= −
=
= −
=
= −
=
d du
DERIV DE FUNCS TRIGO INVER
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
sin
1
1
cos
1
1
tg
1
1
ctg
1
si 11
sec
si 11
si 11
csc
si 11
1
vers
2
u
dx dxu
d du
u
dx dxu
d du
u
dx dxu
d du
u
dx dxu
ud du
u
udx dxu u
ud du
u
udx dxu u
d du
u
dx dxu u
∠ = ⋅
−
∠ = − ⋅
−
∠ = ⋅
+
∠ = − ⋅
+
+ >⎧
∠ = ± ⋅ ⎨
1d du
− < −⎩−
− >⎧
∠ = ⋅ ⎨
+ < −⎩−
∠ = ⋅
−
∓
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
sinh cosh
cosh sinh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
=
=
=
= −
= −
= −
d du
DERIVADA DE FUNCS HIP INV
1
2
-1
1
-12
1
2
1
2
1
1
12
senh
1
si cosh 01
cosh , 1
si cosh 01
1
tgh , 1
1
1
ctgh , 1
1
si sech 0, 0,11
sech
si sech 0, 0,11
u
dx dxu
ud du
u u
dx dx uu
d du
u u
dx u dx
d du
u u
dx u dx
u ud du
u
dx dx u uu u
−
−
−
−
−
−
−
= ⋅
+
⎧+ >± ⎪
= ⋅ > ⎨
− <− ⎪⎩
= ⋅ <
−
= ⋅ >
−
⎧− > ∈⎪
= ⋅ ⎨
+ < ∈− ⎩
∓
1d du
1
2
1
csch , 0
1
d du
u u
dx dxu u
−
⎪
= − ⋅ ≠
+
INTEGRALES DEFINIDAS, PROPIEDADES
Nota. Para todas las fórmulas de integración deberá
agregarse una constante arbitraria c (constante de
integración).
( ) ( ){ } ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) [ ]
( ) ( )
( ) ( ) [ ]
( ) ( )
0
, , ,
,
si
b b b
a a a
b b
a a
b c b
a a c
b a
a b
a
a
b
a
b b
a a
b b
a a
f x g x dx f x dx g x dx
cf x dx c f x dx c
f x dx f x dx f x dx
f x dx f x dx
f x dx
m b a f x dx M b a
m f x M x a b m M
f x dx g x dx
f x g x x a b
f x dx f x dx a b
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
≤ <
∫ ∫ ∫
∫ ∫
∫ ∫ ∫
∫ ∫
∫
∫
∫ ∫
∫ ∫
INTEGRALES
( ) ( )
( )
( )
1
Integración por partes
1
1
ln
n
n
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
u
u du n
n
du
u
u
+
=
=
± ± ± = ± ± ±
= −
= ≠ −
+
=
∫
∫ ∫
∫ ∫ ∫ ∫
∫ ∫
∫
∫
INTEGRALES DE FUNCS LOG & EXP
( )
( )
( ) ( )
( )
( )
2
2
0
1ln
1
ln ln
1
ln ln ln 1
1
log ln ln 1
ln ln
log 2log 1
4
ln 2ln 1
4
u u
u
u
u
u
u u
a
a a
e du e
aa
a du
aa
a
ua du u
a a
ue du e u
udu u u u u u
u
udu u u u u
a a
u
u udu u
u
u udu u
=
>⎧
= ⎨
≠⎩
⎛ ⎞
= ⋅ −⎜ ⎟
⎝ ⎠
= −
= − = −
= − = −
= ⋅ −
= −
∫
∫
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS TRIGO
2
2
sin cos
cos sin
sec tg
csc ctg
sec tg sec
csc ctg csc
udu u
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫
∫
∫
∫
∫
∫
tg ln cos ln sec
ctg ln sin
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫
∫
∫
∫
( )
2
2
2
2
1
sin sin 2
2 4
1
cos sin 2
2 4
tg tg
ctg ctg
u
udu u
u
udu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫
∫
∫
sin sin cos
cos cos sin
u udu u u u
u udu u u u
= −
= +
∫
∫
INTEGRALES DE FUNCS TRIGO INV
( )
( )
2
2
2
2
2
2
sin sin 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
csc cosh
udu u u u
udu u u u
udu u u u
udu u u u
udu u u u u
u u u
udu u u u u
u u u
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − +
= ∠ − ∠
∠ = ∠ + + −
= ∠ + ∠
∫
∫
∫
∫
∫
∫
−
INTEGRALES DE FUNCS HIP
2
2
sinh cosh
cosh sinh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu u
udu u
udu u
udu u
u udu u
u udu u
=
=
=
= −
= −
= −
∫
∫
∫
∫
∫
∫
( )
( )1
tgh lncosh
ctgh ln sinh
sech tg sinh
csch ctgh cosh
1
ln tgh
2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫
∫
∫
∫
INTEGRALES DE FRAC
( )
( )
2 2
2 2
2 2
2 2
2 2
tg
1
ctg
1
ln
2
1
ln
2
du
u a a a
u
a a
du u a
u a
u a a u a
du a u
u a
a u a a u
= ∠
+
= − ∠
−
= >
− +
+
= <
− −
∫
∫
∫
1 u
INTEGRALES CON
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
2
2 2 2 2
2
2 2 2 2 2 2
sin
cos
ln
1
ln
1
cos
1
sec
sen
2 2
ln
2 2
du u
aa u
u
a
du
u u a
u a
du u
au a u a a u
du a
a uu u a
u
a a
u a u
a u du a u
a
u a
u a du u a u u a
= ∠
−
= −∠
= + ±
±
=
± + ±
= ∠
−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
MÁS INTEGRALES
( )
( )
2 2
2 2
3
sin cos
sin
cos sin
cos
1 1
sec sec tg ln sec tg
2 2
au
au
au
e a bu b bu
e bu du
a b
e a bu b bu
e bu du
a b
udu u u u u
−
=
+
+
=
+
= + +
∫
∫
∫
au
ALGUNAS SERIES
( ) ( ) ( )( )
( )( )
( )
( )( )
( ) ( ) ( )
( )
( )
( )
( )
( )
2
0 0
0 0 0
0 0
2
2 3
3 5 7 2 1
1
2 4 6
''
'
2!
: Taylor
!
'' 0
0 ' 0
2!
0
: Maclaurin
!
1
2! 3! !
sin 1
3! 5! 7! 2 1 !
cos 1
2! 4!
nn
n n
n
x
n
n
f x x x
f x f x f x x x
f x x x
n
f x
f x f f x
f x
n
x x x
e x
n
x x x x
x x
n
x x x
x
−
−
−
= + − +
−
+ +
= + +
+ +
= + + + + + +
= − + − + + −
−
= − + − ( )
( )
( ) ( )
( )
2 2
1
2 3 4
1
3 5 7 2 1
1
1
6! 2 2 !
ln 1 1
2 3 4
tg 1
3 5 7 2 1
n
n
n
n
n
n
x
n
x x x x
x x
n
x x x x
x x
n
−
−
−
−
−
+ + −
−
+ = − + − + + −
∠ = − + − + + −
−
Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
ALFABETO GRIEGO
ayúscula Minúscula NombreM Equivalente
Romano
1 Α α Alfa A
2 Β β Beta B
3 Γ γ Gamma G
4 ∆ δ Delta D
5 Ε ε Epsilon E
6 Ζ ζ Zeta Z
7 Η η Eta H
8 Θ θ ϑ Teta Q
9 Ι ι Iota I
10 Κ κ Kappa K
11 Λ λ Lambda L
12 Μ µ Mu M
13 Ν ν Nu N
14 Ξ ξ Xi X
15 Ο ο Omicron O
16 Π π ϖ Pi P
17 Ρ ρ Rho R
18 Σ σ ς Sigma S
19 Τ τ Tau T
20 Υ υ Ipsilon U
21 Φ φ ϕ Phi F
22 Χ χ Ji C
23 Ψ ψ Psi Y
24 Ω ω Omega W
NOTACIÓN
Seno.sin
cos Coseno.
tg Tangente.
sec Secante.
csc Cosecante.
ctg Cotangente.
vers Verso seno.
arcsin sinθ θ= Arco seno de un ángulo θ .
( )u f x=
sinh Seno hiperbólico.
cosh Coseno hiperbólico.
tgh Tangente hiperbólica.
ctgh Cotangente hiperbólica.
sech Secante hiperbólica.
csch Cosecante hiperbólica.
, ,u v w Funciones de x , , .( )u u x= ( )v v x=
Conjunto de los números reales.
{ } Conjunto de enteros., 2, 1,0,1,2,= − −… …
Conjunto de números racionales.
c
Conjunto de números irracionales.
{ }1,2,3,= … Conjunto de números naturales.
Conjunto de números complejos.

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Formulario

  • 1. Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. Fórmulas de Cálculo Diferencial e Integral VER.6.8 Jesús Rubí Miranda (jesusrubim@yahoo.com) http://www.geocities.com/calculusjrm/ VALOR ABSOLUTO 1 1 1 1 si 0 si 0 y 0 y 0 0 ó ó n n k k k k n n k k k k a a a a a a a a a a a a a a ab a b a a a b a b a a = = = = ≥⎧ = ⎨ − <⎩ = − ≤ − ≤ ≥ = ⇔ = = = + ≤ + ≤ ∏ ∏ ∑ ∑ EXPONENTES ( ) ( ) / p q p q p p q q qp pq p p p p p p qp q p a a a a a a a a a b a b a a b b a a + − ⋅ = = = ⋅ = ⋅ ⎛ ⎞ =⎜ ⎟ ⎝ ⎠ = LOGARITMOS 10 log log log log log log log log log log ln log log ln log log y log ln x a a a a a a a r a a b a b e N x a MN M N M M N N N r N N N N a a N N N N N = ⇒ = + = − = = = = = = ALGUNOS PRODUCTOS ad+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 3 3 2 2 3 3 3 2 2 3 2 2 2 2 2 2 3 3 3 3 2 2 2 a c d ac a b a b a b a b a b a b a ab b a b a b a b a ab b x b x d x b d x bd ax b cx d acx ad bc x bd a b c d ac ad bc bd a b a a b ab b a b a a b ab b a b c a b c ab ac bc ⋅ + = + ⋅ − = − + ⋅ + = + = + + − ⋅ − = − = − + + ⋅ + = + + + + ⋅ + = + + + + ⋅ + = + + + + = + + + − = − + − + + = + + + + + 1 1 n n k k n n k a b a ab b a b a b a a b ab b a b a b a a b a b ab b a b a b a b a b n− − = − ⋅ + + = − − ⋅ + + + = − − ⋅ + + + + = − ⎛ ⎞ − ⋅ = − ∀ ∈⎜ ⎟ ⎝ ⎠ ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 5 4 3 2 2 3 4 5 6 6 a b a ab b a b a b a a b ab b a b a b a a b a b ab b a b a b a a b a b a b ab b a b + ⋅ − + = + + ⋅ − + − = − + ⋅ − + − + = + + ⋅ − + − + − = − ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 impar 1 par n k n k k n n k n k n k k n n k a b a b a b n a b a b a b n + − − = + − − = ⎛ ⎞ + ⋅ − = + ∀ ∈⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + ⋅ − = − ∀ ∈⎜ ⎟ ⎝ ⎠ ∑ ∑ SUMAS Y PRODUCTOS n ( ) ( ) 1 2 1 1 1 1 1 1 1 1 0 n k k n k n n k k k k n n n k k k k k k k n k k n k a a a a c nc ca c a a b a b a a a a = = = = = = = − = + + + = = = + = + − = − ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 2 3 2 1 3 4 3 2 1 4 5 4 3 1 2 1 1 2 1 2 = 2 1 1 1 1 2 1 2 3 6 1 2 4 1 6 15 10 30 1 3 5 2 1 ! n k nn k k n k n k n k n k n k n a k d a n d n a l r a rl ar a r r k n n k n n n k n n n k n n n n n n n k n n k = − = = = = = = + − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ + − − = = − − = + = + + = + + = + + − + + + + − = = ⎛ ⎞ =⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ ∑ ∑ ∑ ∏ ( ) ( ) 0 ! , ! ! n n n k k k k n n k k n x y x y k − = ≤ − ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ ∑ ( ) 1 2 1 2 1 2 1 2 ! ! ! ! k n nn n k k k n x x x x x x n n n + + + = ⋅∑ CONSTANTES 9…3.1415926535 2.71828182846e π = = … TRIGONOMETRÍA 1 sen csc sen 1 cos sec cos sen 1 tg ctg cos tg CO HIP CA HIP CO CA θ θ θ θ θ θ θ θ θ θ θ = = = = = = = radianes=180π CA CO HIP θ θ sin cos tg ctg sec csc 0 0 1 0 ∞ 1 ∞ 30 1 2 3 2 1 3 3 2 3 2 45 1 2 1 2 1 1 2 2 60 3 2 1 2 3 1 3 2 2 3 90 1 0 ∞ 0 ∞ 1 [ ] [ ] sin , 2 2 cos 0, tg , 2 2 1 ctg tg 0, 1 sec cos 0, 1 csc sen , 2 2 y x y y x y y x y y x y x y x y x y x y x π π π π π π π π π ⎡ ⎤ = ∠ ∈ −⎢ ⎥ ⎣ ⎦ = ∠ ∈ = ∠ ∈ − = ∠ = ∠ ∈ = ∠ = ∠ ∈ ⎡ ⎤ = ∠ = ∠ ∈ −⎢ ⎥ ⎣ ⎦ Gráfica 1. Las funciones trigonométricas: sin x , cos x , tg x : -8 -6 -4 -2 0 2 4 6 8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 sen x cos x tg x Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x : -8 -6 -4 -2 0 2 4 6 8 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 csc x sec x ctg x Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctg x : -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 arc sen x arc cos x arc tg x Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : -5 0 5 -2 -1 0 1 2 3 4 arc ctg x arc sec x arc csc x IDENTIDADES TRIGONOMÉTRICAS 2 2 2 2 2 2 sin cos 1 1 ctg csc tg 1 sec θ θ θ θ θ θ + = + = + = ( ) ( ) ( ) sin sin cos cos tg tg θ θ θ θ θ θ − = − − = − = − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin 2 sin cos 2 cos tg 2 tg sin sin cos cos tg tg sin 1 sin cos 1 cos tg tg n n n n n θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ + = + = + = + = − + = − + = + = − + = − + = ( ) ( ) ( ) ( ) ( ) sin 0 cos 1 tg 0 2 1 sin 1 2 2 1 cos 0 2 2 1 tg 2 n n n n n n n n π π π π π π = = − = +⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ +⎛ ⎞ =⎜ ⎟ ⎝ ⎠ +⎛ ⎞ = ∞⎜ ⎟ ⎝ ⎠ sin cos 2 cos sin 2 π θ θ π θ θ ⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = +⎜ ⎟ ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 sin sin cos cos sin cos cos cos sin sin tg tg tg 1 tg tg sin 2 2sin cos cos2 cos sin 2tg tg2 1 tg 1 sin 1 cos2 2 1 cos 1 cos2 2 1 cos2 tg 1 cos2 α β α β α β α β α β α β α β α β α β θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ± = ± ± = ± ± = = = − = − = − = + − = + ∓ ∓ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 sin sin 2sin cos 2 2 1 1 sin sin 2sin cos 2 2 1 1 cos cos 2cos cos 2 2 1 1 cos cos 2sin sin 2 2 α β α β α β α β α β α β α β α β α β α β α β α β + = + ⋅ − − = − ⋅ + + = + ⋅ − − = − + ⋅ − ( )sin tg tg cos cos α β α β α β ± ± = ⋅ ( ) ( ) ( ) ( ) ( ) ( ) 1 sin cos sin sin 2 1 sin sin cos cos 2 1 cos cos cos cos 2 α β α β α β α β α β α β α β α β α β ⋅ = − + +⎡ ⎤⎣ ⎦ ⋅ = − − +⎡ ⎤⎣ ⎦ ⋅ = − + +⎡ ⎤⎣ ⎦ tg tg tg tg ctg ctg α β α β α β + ⋅ = + FUNCIONES HIPERBÓLICAS sinh 2 cosh 2 sinh tgh cosh 1 ctgh tgh 1 2 sech cosh 1 2 csch sinh x x x x x x x x x x x x x e e x x e e x x e e e e x x e e x x e e x x e e − − − − − − − = + = − = = + + = = − = = + = = − x x e e− − x x [ { } ] { } { } sinh : cosh : 1, tgh : 1,1 ctgh : 0 , 1 1, sech : 0,1 csch : 0 0 → → ∞ → − − → −∞ − ∪ ∞ → − → − Gráfica 5. Las funciones hiperbólicas sinh x , cosh x , tgh x : -5 0 5 -4 -3 -2 -1 0 1 2 3 4 5 senh x cosh x tgh x FUNCIONES HIPERBÓLICAS INV ( ) ( ) 1 2 1 2 1 1 2 1 2 1 sinh ln 1 , cosh ln 1 , 1 1 1 tgh ln , 1 2 1 1 1 ctgh ln , 1 2 1 1 1 sech ln , 0 1 1 1 csch ln , 0 x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − − = + + ∀ ∈ = ± − ≥ +⎛ ⎞ = <⎜ ⎟ −⎝ ⎠ +⎛ ⎞ = >⎜ ⎟ −⎝ ⎠ ⎛ ⎞± − ⎜ ⎟= < ≤ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞+ ⎜ ⎟= + ≠ ⎜ ⎟ ⎝ ⎠
  • 2. Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. IDENTIDADES DE FUNCS HIP 2 2 sinh 1x x ( ) ( ) ( ) 2 2 2 2 cosh 1 tgh sech ctgh 1 csch sinh sinh cosh cosh tgh tgh x x x x x x x x x x − = − = − = − − = − = − − = ( ) ( ) ( ) 2 2 2 sinh sinh cosh cosh sinh cosh cosh cosh sinh sinh tgh tgh tgh 1 tgh tgh sinh 2 2sinh cosh cosh 2 cosh sinh 2tgh tgh 2 1 tgh x y x y x y x y x y x y x y x y x y x x x x x x x x x ± = ± ± = ± ± ± = ± = = + = + ( ) ( ) 2 2 2 1 sinh cosh 2 1 2 1 cosh cosh 2 1 2 cosh 2 1 tgh cosh 2 1 x x x x x x x = − = + − = + sinh 2 tgh cosh 2 1 x x x = + cosh sinh cosh sinh x x e x x e x x− = + = − OTRAS ( ) ( ) 2 2 2 0 4 2 4 discriminante exp cos sin si , ax bx c b b ac x a b ac i e iα α β β β α β + + = − ± − ⇒ = − = ± = ± ∈ LÍMITES ( ) 1 0 0 0 0 1 lim 1 2.71828... 1 lim 1 sen lim 1 1 cos lim 0 1 lim 1 1 lim 1 ln x x x x x x x x x x e e x x x x x e x x x → →∞ → → → → + = = ⎛ ⎞ + =⎜ ⎟ ⎝ ⎠ = − = − = − = DERIVADAS ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 lim lim 0 x x x n n f x x f xdf y D f x dx x x d c dx d cx c dx d cx ncx dx d du dv dw u v w dx dx dx dx d du cu c dx dx ∆ → ∆ → − + ∆ − ∆ = = = ∆ ∆ = = = ± ± ± = ± ± ± = ( ) ( ) ( ) ( ) ( ) 2 1n n d dv du uv u v dx dx dx d dw dv du uvw uv uw vw dx dx dx dx v du dx u dv dxd u dx v v d du u nu dx dx − = + = + + −⎛ ⎞ =⎜ ⎟ ⎝ ⎠ = ( ) ( ) ( ) ( ) 12 1 2 (Regla de la Cadena) 1 donde dF dF du dx du dx du dx dx du dF dudF dx dx du x f tf tdy dtdy dx dx dt f t y f t = ⋅ = = =⎧′ ⎪ = = ⎨ ′ =⎪⎩ DERIVADA DE FUNCS LOG & EXP ( ) ( ) ( ) ( ) ( ) ( ) 1 ln log log log log 0, 1 ln ln a a u u u u v v v u dx u u dx d e du u dx u dx ed du u a dx u dx d du e e dx dx d du a a a dx dx d du dv u vu u u dx dx dx − = = ⋅ = ⋅ = ⋅ > = ⋅ = ⋅ = + ⋅ ⋅ 1d du dx du a ≠ DERIVADA DE FUNCIONES TRIGO ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 sin cos cos sin tg sec ctg csc sec sec tg csc csc ctg vers sen u u dx dx d du u u dx dx d du u u dx dx d du u u dx dx d du u u u dx dx d du u u u dx dx d du u u dx dx = = − = = − = = − = d du DERIV DE FUNCS TRIGO INVER ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 sin 1 1 cos 1 1 tg 1 1 ctg 1 si 11 sec si 11 si 11 csc si 11 1 vers 2 u dx dxu d du u dx dxu d du u dx dxu d du u dx dxu ud du u udx dxu u ud du u udx dxu u d du u dx dxu u ∠ = ⋅ − ∠ = − ⋅ − ∠ = ⋅ + ∠ = − ⋅ + + >⎧ ∠ = ± ⋅ ⎨ 1d du − < −⎩− − >⎧ ∠ = ⋅ ⎨ + < −⎩− ∠ = ⋅ − ∓ DERIVADA DE FUNCS HIPERBÓLICAS 2 2 sinh cosh cosh sinh tgh sech ctgh csch sech sech tgh csch csch ctgh u u dx dx d du u u dx dx d du u u dx dx d du u u dx dx d du u u u dx dx d du u u u dx dx = = = = − = − = − d du DERIVADA DE FUNCS HIP INV 1 2 -1 1 -12 1 2 1 2 1 1 12 senh 1 si cosh 01 cosh , 1 si cosh 01 1 tgh , 1 1 1 ctgh , 1 1 si sech 0, 0,11 sech si sech 0, 0,11 u dx dxu ud du u u dx dx uu d du u u dx u dx d du u u dx u dx u ud du u dx dx u uu u − − − − − − − = ⋅ + ⎧+ >± ⎪ = ⋅ > ⎨ − <− ⎪⎩ = ⋅ < − = ⋅ > − ⎧− > ∈⎪ = ⋅ ⎨ + < ∈− ⎩ ∓ 1d du 1 2 1 csch , 0 1 d du u u dx dxu u − ⎪ = − ⋅ ≠ + INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de integración). ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) 0 , , , , si b b b a a a b b a a b c b a a c b a a b a a b a b b a a b b a a f x g x dx f x dx g x dx cf x dx c f x dx c f x dx f x dx f x dx f x dx f x dx f x dx m b a f x dx M b a m f x M x a b m M f x dx g x dx f x g x x a b f x dx f x dx a b ± = ± = ⋅ ∈ = + = − = ⋅ − ≤ ≤ ⋅ − ⇔ ≤ ≤ ∀ ∈ ∈ ≤ ⇔ ≤ ∀ ∈ ≤ < ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES ( ) ( ) ( ) ( ) 1 Integración por partes 1 1 ln n n adx ax af x dx a f x dx u v w dx udx vdx wdx udv uv vdu u u du n n du u u + = = ± ± ± = ± ± ± = − = ≠ − + = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES DE FUNCS LOG & EXP ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 1ln 1 ln ln 1 ln ln ln 1 1 log ln ln 1 ln ln log 2log 1 4 ln 2ln 1 4 u u u u u u u u a a a e du e aa a du aa a ua du u a a ue du e u udu u u u u u u udu u u u u a a u u udu u u u udu u = >⎧ = ⎨ ≠⎩ ⎛ ⎞ = ⋅ −⎜ ⎟ ⎝ ⎠ = − = − = − = − = − = ⋅ − = − ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES DE FUNCS TRIGO 2 2 sin cos cos sin sec tg csc ctg sec tg sec csc ctg csc udu u udu u udu u udu u u udu u u udu u = − = = = − = = − ∫ ∫ ∫ ∫ ∫ ∫ tg ln cos ln sec ctg ln sin sec ln sec tg csc ln csc ctg udu u u udu u udu u u udu u u = − = = = + = − ∫ ∫ ∫ ∫ ( ) 2 2 2 2 1 sin sin 2 2 4 1 cos sin 2 2 4 tg tg ctg ctg u udu u u udu u udu u u udu u u = − = + = − = − + ∫ ∫ ∫ ∫ sin sin cos cos cos sin u udu u u u u udu u u u = − = + ∫ ∫ INTEGRALES DE FUNCS TRIGO INV ( ) ( ) 2 2 2 2 2 2 sin sin 1 cos cos 1 tg tg ln 1 ctg ctg ln 1 sec sec ln 1 sec cosh csc csc ln 1 csc cosh udu u u u udu u u u udu u u u udu u u u udu u u u u u u u udu u u u u u u u ∠ = ∠ + − ∠ = ∠ − − ∠ = ∠ − + ∠ = ∠ + + ∠ = ∠ − + = ∠ − ∠ ∠ = ∠ + + − = ∠ + ∠ ∫ ∫ ∫ ∫ ∫ ∫ − INTEGRALES DE FUNCS HIP 2 2 sinh cosh cosh sinh sech tgh csch ctgh sech tgh sech csch ctgh csch udu u udu u udu u udu u u udu u u udu u = = = = − = − = − ∫ ∫ ∫ ∫ ∫ ∫ ( ) ( )1 tgh lncosh ctgh ln sinh sech tg sinh csch ctgh cosh 1 ln tgh 2 udu u udu u udu u udu u u − = = = ∠ = − = ∫ ∫ ∫ ∫ INTEGRALES DE FRAC ( ) ( ) 2 2 2 2 2 2 2 2 2 2 tg 1 ctg 1 ln 2 1 ln 2 du u a a a u a a du u a u a u a a u a du a u u a a u a a u = ∠ + = − ∠ − = > − + + = < − − ∫ ∫ ∫ 1 u INTEGRALES CON ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin cos ln 1 ln 1 cos 1 sec sen 2 2 ln 2 2 du u aa u u a du u u a u a du u au a u a a u du a a uu u a u a a u a u a u du a u a u a u a du u a u u a = ∠ − = −∠ = + ± ± = ± + ± = ∠ − = ∠ − = − + ∠ ± = ± ± + ± ∫ ∫ ∫ ∫ ∫ ∫ MÁS INTEGRALES ( ) ( ) 2 2 2 2 3 sin cos sin cos sin cos 1 1 sec sec tg ln sec tg 2 2 au au au e a bu b bu e bu du a b e a bu b bu e bu du a b udu u u u u − = + + = + = + + ∫ ∫ ∫ au ALGUNAS SERIES ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 0 0 0 0 0 0 2 2 3 3 5 7 2 1 1 2 4 6 '' ' 2! : Taylor ! '' 0 0 ' 0 2! 0 : Maclaurin ! 1 2! 3! ! sin 1 3! 5! 7! 2 1 ! cos 1 2! 4! nn n n n x n n f x x x f x f x f x x x f x x x n f x f x f f x f x n x x x e x n x x x x x x n x x x x − − − = + − + − + + = + + + + = + + + + + + = − + − + + − − = − + − ( ) ( ) ( ) ( ) ( ) 2 2 1 2 3 4 1 3 5 7 2 1 1 1 6! 2 2 ! ln 1 1 2 3 4 tg 1 3 5 7 2 1 n n n n n n x n x x x x x x n x x x x x x n − − − − − + + − − + = − + − + + − ∠ = − + − + + − −
  • 3. Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. ALFABETO GRIEGO ayúscula Minúscula NombreM Equivalente Romano 1 Α α Alfa A 2 Β β Beta B 3 Γ γ Gamma G 4 ∆ δ Delta D 5 Ε ε Epsilon E 6 Ζ ζ Zeta Z 7 Η η Eta H 8 Θ θ ϑ Teta Q 9 Ι ι Iota I 10 Κ κ Kappa K 11 Λ λ Lambda L 12 Μ µ Mu M 13 Ν ν Nu N 14 Ξ ξ Xi X 15 Ο ο Omicron O 16 Π π ϖ Pi P 17 Ρ ρ Rho R 18 Σ σ ς Sigma S 19 Τ τ Tau T 20 Υ υ Ipsilon U 21 Φ φ ϕ Phi F 22 Χ χ Ji C 23 Ψ ψ Psi Y 24 Ω ω Omega W NOTACIÓN Seno.sin cos Coseno. tg Tangente. sec Secante. csc Cosecante. ctg Cotangente. vers Verso seno. arcsin sinθ θ= Arco seno de un ángulo θ . ( )u f x= sinh Seno hiperbólico. cosh Coseno hiperbólico. tgh Tangente hiperbólica. ctgh Cotangente hiperbólica. sech Secante hiperbólica. csch Cosecante hiperbólica. , ,u v w Funciones de x , , .( )u u x= ( )v v x= Conjunto de los números reales. { } Conjunto de enteros., 2, 1,0,1,2,= − −… … Conjunto de números racionales. c Conjunto de números irracionales. { }1,2,3,= … Conjunto de números naturales. Conjunto de números complejos.