Fórmulas de Cálculo Diferencial e Integral (Jesús Rubí M.)

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Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3)
Fórmulas de
Cálculo Diferencial
e Integral VER.6.8
Jesús Rubí Miranda ([email protected])
http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO
( a + b ) ⋅ ( a 2 − ab + b 2 ) = a 3 + b3
( a + b ) ⋅ ( a3 − a 2 b + ab 2 − b3 ) = a 4 − b 4
( a + b ) ⋅ ( a 4 − a 3b + a 2 b 2 − ab3 + b 4 ) = a 5 + b5
( a + b ) ⋅ ( a5 − a 4 b + a 3b 2 − a 2 b3 + ab 4 − b5 ) = a 6 − b 6
⎛ n
⎞
k +1
( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈
⎝ k =1
⎠
⎛
⎞
a n − k b k −1 ⎟ = a n − b n ∀ n ∈
⎝ k =1
⎠
SUMAS Y PRODUCTOS
a = −a
a1 + a2 +
a ≤ a y −a ≤ a
a ≥0 y a =0 ⇔ a=0
ab = a b ó
n
a+b ≤ a + b ó
k
n
n
≤ ∑ ak
k
k =1
n
k =1
ap
= a p−q
aq
(a ⋅b)
=a
p
k =1
k =1
∑(a
n
k =1
k =1
= a ⋅b
p
p
ap
⎛a⎞
⎜ ⎟ = p
b
⎝b⎠
a p/q = a p
q
LOGARITMOS
log a N = x ⇒ a x = N
log a MN = log a M + log a N
M
= log a M − log a N
N
log a N r = r log a N
log a
log b N ln N
=
log a N =
log b a ln a
1+ 3 + 5 +
y∈ −
π π
,
2 2
log10 N = log N y log e N = ln N
n! = ∏ k
ALGUNOS PRODUCTOS
a ⋅ ( c + d ) = ac + ad
( a + b) ⋅ ( a − b) = a − b
2
( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2
2
( a − b ) ⋅ ( a − b ) = ( a − b ) = a 2 − 2ab + b 2
( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd
( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd
( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd
3
( a + b ) = a3 + 3a 2b + 3ab 2 + b3
3
( a − b ) = a 3 − 3a 2b + 3ab 2 − b3
2
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
2
2
( a − b ) ⋅ ( a + ab + b ) = a − b
( a − b ) ⋅ ( a 3 + a 2 b + ab 2 + b3 ) = a 4 − b 4
( a − b ) ⋅ ( a 4 + a 3b + a 2 b 2 + ab3 + b 4 ) = a 5 − b5
2
n
2
⎞
3
3
( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n
⎝ k =1
⎠
∀n ∈
tg (θ + π ) = tg θ
sen x
cos x
tg x
-1.5
-2
0
2
4
6
sin (θ + nπ ) = ( −1) sin θ
n
8
Gráfica 2. Las funciones trigonométricas csc x ,
sec x , ctg x :
2.5
1
0.5
2
0
-1.5
-2
0
2
4
6
8
xknk
4
n
⎛ 2n + 1 ⎞
sin ⎜
π ⎟ = ( −1)
⎝ 2
⎠
⎛ 2n + 1 ⎞
cos ⎜
π⎟=0
⎝ 2
⎠
⎛ 2n + 1 ⎞
tg ⎜
π⎟=∞
⎝ 2
⎠
cosh :
tgh :
ctgh :
→
→ [1, ∞
→ −1,1
− {0} → −∞ , −1 ∪ 1, ∞
sech :
→ 0 ,1]
csch :
− {0} →
-1
arc sen x
arc cos x
arc tg x
-2
-1
0
1
2
3
cos 2θ = cos 2 θ − sin 2 θ
2 tg θ
tg 2θ =
1 − tg 2 θ
1
sin 2 θ = (1 − cos 2θ )
2
1
cos 2 θ = (1 + cos 2θ )
2
1 − cos 2θ
tg 2 θ =
1 + cos 2θ
− {0}
Gráfica 5. Las funciones hiperbólicas sinh x ,
cosh x , tgh x :
5
4
π⎞
⎛
sin θ = cos ⎜ θ − ⎟
2⎠
⎝
π⎞
⎛
cos θ = sin ⎜ θ + ⎟
2⎠
⎝
tg α ± tg β
tg (α ± β ) =
1 ∓ tg α tg β
sin 2θ = 2sin θ cos θ
0
CO
sinh :
n
3
2
1
0
-1
-2
cos (α ± β ) = cos α cos β ∓ sin α sin β
1
-2
-3
tg (θ + nπ ) = tg θ
sin ( nπ ) = 0
sin (α ± β ) = sin α cos β ± cos α sin β
2
π radianes=180
CA
-4
3
e = 2.71828182846…
TRIGONOMETRÍA
CO
1
sen θ =
cscθ =
HIP
sen θ
CA
1
cosθ =
secθ =
HIP
cosθ
sen θ CO
1
tgθ =
ctgθ =
=
cosθ CA
tgθ
θ
-6
Gráfica 3. Las funciones trigonométricas inversas
arcsin x , arccos x , arctg x :
CONSTANTES
π = 3.14159265359…
HIP
csc x
sec x
ctg x
-2
n!
x1n1 ⋅ x2n2
n1 ! n2 ! nk !
n
tg ( nπ ) = 0
1.5
-1
n
cos (θ + nπ ) = ( −1) cos θ
cos ( nπ ) = ( −1)
2
ex − e− x
2
e x + e− x
cosh x =
2
sinh x e x − e − x
=
tgh x =
cosh x e x + e− x
e x + e− x
1
=
ctgh x =
tgh x e x − e − x
1
2
=
sech x =
cosh x e x + e − x
1
2
=
csch x =
sinh x e x − e − x
sinh x =
cos (θ + π ) = − cos θ
-0.5
+ xk ) = ∑
tg ( −θ ) = − tg θ
tg (θ + 2π ) = tg θ
-1
sin α ⋅ cos β =
tg α + tg β
ctg α + ctg β
FUNCIONES HIPERBÓLICAS
sin (θ + π ) = − sin θ
-4
5
cos ( −θ ) = cos θ
cos (θ + 2π ) = cos θ
k =1
( x1 + x2 +
tg α ⋅ tg β =
2
-0.5
-2.5
-8
sin ( −θ ) = − sin θ
sin θ + cos 2 θ = 1
1 + ctg 2 θ = csc 2 θ
0
⎛n⎞
n!
, k≤n
⎜ ⎟=
⎝ k ⎠ ( n − k )!k !
n
⎛n⎞
n
( x + y ) = ∑ ⎜ ⎟ xn−k y k
k =0 ⎝ k ⎠
tg 2 θ + 1 = sec 2 θ
y ∈ 0, π
sin (θ + 2π ) = sin θ
-6
sin (α ± β )
cos α ⋅ cos β
1
⎡sin (α − β ) + sin (α + β ) ⎦⎤
2⎣
1
sin α ⋅ sin β = ⎣⎡cos (α − β ) − cos (α + β ) ⎦⎤
2
1
cos α ⋅ cos β = ⎣⎡cos (α − β ) + cos (α + β ) ⎦⎤
2
0
IDENTIDADES TRIGONOMÉTRICAS
0.5
n
tg α ± tg β =
arc ctg x
arc sec x
arc csc x
-2
-5
1
-2
-8
Jesús Rubí M.
1
1
(α + β ) ⋅ cos (α − β )
2
2
1
1
sin α − sin β = 2 sin (α − β ) ⋅ cos (α + β )
2
2
1
1
cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β )
2
2
1
1
cos α − cos β = −2 sin (α + β ) ⋅ sin (α − β )
2
2
sin α + sin β = 2sin
0
2
+ ( 2n − 1) = n
2
-1
1.5
pq
3
1
Gráfica 1. Las funciones trigonométricas: sin x ,
cos x , tg x :
n
∑
⎣⎡ a + ( k − 1) d ⎦⎤ = 2 ⎣⎡ 2a + ( n − 1) d ⎦⎤
k =1
n
= (a + l )
2
n
n
a − rl
1− r
ar k −1 = a
=
∑
1− r
1− r
k =1
n
1 2
k
n
n
=
+
(
)
∑
2
k =1
n
1
2
k = ( 2n3 + 3n 2 + n )
∑
6
k =1
n
1
k 3 = ( n 4 + 2n3 + n 2 )
∑
4
k =1
n
1
4
k
=
( 6n5 + 15n4 + 10n3 − n )
∑
30
k =1
4
y ∈ ⎢− , ⎥
⎣ 2 2⎦
y = ∠ cos x y ∈ [ 0, π ]
n
p
⎛
n
− ak −1 ) = an − a0
k
y = ∠ sin x
1
y = ∠ sec x = ∠ cos
y ∈ [ 0, π ]
x
1
⎡ π π⎤
y = ∠ csc x = ∠ sen
y ∈ ⎢− , ⎥
x
⎣ 2 2⎦
∑ ( ak + bk ) = ∑ ak + ∑ bk
EXPONENTES
(a )
k =1
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x :
⎡ π π⎤
1
y = ∠ ctg x = ∠ tg
x
= c ∑ ak
n
k =1
a p ⋅ a q = a p+q
p q
+ an = ∑ ak
n
k
k =1
tg
ctg
cos
sec
csc
sin
0
0
∞
∞
1
1
12
3 2 3 2
3 2 1 3
1 2 1 2
2
2
1
1
3 1 3
2 2 3
3 2 12
0
0
∞
∞
1
1
y = ∠ tg x
n
k =1
∑ ca
k =1
∑a
par
∑ c = nc
= ∏ ak
n
k +1
n
n
∏a
k =1
n
( a + b ) ⋅ ⎜ ∑ ( −1)
⎧a si a ≥ 0
a =⎨
⎩− a si a < 0
impar
θ
0
30
45
60
90
senh x
cosh x
tgh x
-3
-4
-5
0
5
FUNCIONES HIPERBÓLICAS INV
(
(
)
)
sinh −1 x = ln x + x 2 + 1 , ∀x ∈
cosh −1 x = ln x ± x 2 − 1 , x ≥ 1
tgh −1 x =
1 ⎛1+ x ⎞
ln ⎜
⎟,
2 ⎝1− x ⎠
1 ⎛ x +1⎞
ctgh −1 x = ln ⎜
⎟,
2 ⎝ x −1⎠
x <1
x >1
⎛ 1 ± 1 − x2 ⎞
⎟, 0 < x ≤ 1
sech −1 x = ln ⎜
⎜
⎟
x
⎝
⎠
⎛1
x2 + 1 ⎞
⎟, x ≠ 0
csch −1 x = ln ⎜ +
⎜x
x ⎟⎠
⎝
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Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3)
IDENTIDADES DE FUNCS HIP
cosh 2 x − sinh 2 x = 1
1 − tgh 2 x = sech 2 x
ctgh 2 x − 1 = csch 2 x
sinh ( − x ) = − sinh x
cosh ( − x ) = cosh x
tgh ( − x ) = − tgh x
sinh ( x ± y ) = sinh x cosh y ± cosh x sinh y
cosh ( x ± y ) = cosh x cosh y ± sinh x sinh y
tgh x ± tgh y
1 ± tgh x tgh y
sinh 2 x = 2sinh x cosh x
tgh ( x ± y ) =
cosh 2 x = cosh 2 x + sinh 2 x
2 tgh x
tgh 2 x =
1 + tgh 2 x
1
sinh 2 x = ( cosh 2 x − 1)
2
1
cosh 2 x = ( cosh 2 x + 1)
2
cosh 2 x − 1
tgh 2 x =
cosh 2 x + 1
sinh 2 x
tgh x =
cosh 2 x + 1
e x = cosh x + sinh x
e − x = cosh x − sinh x
OTRAS
ax 2 + bx + c = 0
−b ± b 2 − 4ac
2a
b 2 − 4ac = discriminante
⇒ x=
exp (α ± i β ) = e
α
( cos β ± i sin β )
si α , β ∈
LÍMITES
1
lim (1 + x ) x = e = 2.71828...
x →0
x
⎛ 1⎞
lim ⎜1 + ⎟ = e
x →∞
⎝ x⎠
sen x
lim
=1
x →0
x
1 − cos x
lim
=0
x →0
x
ex −1
lim
=1
x →0
x
x −1
lim
=1
x →1 ln x
DERIVADAS
f ( x + ∆x ) − f ( x )
df
∆y
Dx f ( x ) =
= lim
= lim
∆x → 0 ∆x
dx ∆x →0
∆x
d
(c) = 0
dx
d
( cx ) = c
dx
d
( cx n ) = ncx n−1
dx
d
du dv dw
(u ± v ± w ± ) = ± ± ±
dx
dx dx dx
d
du
( cu ) = c
dx
dx
d
dv
du
( uv ) = u + v
dx
dx
dx
d
dw
dv
du
( uvw ) = uv + uw + vw
dx
dx
dx
dx
d ⎛ u ⎞ v ( du dx ) − u ( dv dx )
⎜ ⎟=
dx ⎝ v ⎠
v2
d n
n −1 du
( u ) = nu dx
dx
dF dF du
=
⋅
(Regla de la Cadena)
dx du dx
du
1
=
dx dx du
dF dF du
=
dx dx du
dy dy dt f 2′ ( t )
⎪⎧ x = f1 ( t )
=
=
donde ⎨
dx dx dt
f1′( t )
⎪⎩ y = f 2 ( t )
DERIVADA DE FUNCS LOG & EXP
d
du dx 1 du
= ⋅
( ln u ) =
dx
u
u dx
d
log e du
⋅
( log u ) =
dx
u dx
log e du
d
( log a u ) = a ⋅ a > 0, a ≠ 1
dx
u
dx
d u
du
e ) = eu ⋅
(
dx
dx
d u
du
a ) = a u ln a ⋅
(
dx
dx
d v
du
dv
+ ln u ⋅ u v ⋅
u ) = vu v −1
(
dx
dx
dx
DERIVADA DE FUNCIONES TRIGO
d
du
( sin u ) = cos u
dx
dx
d
du
( cos u ) = − sin u
dx
dx
d
du
( tg u ) = sec2 u
dx
dx
d
du
( ctg u ) = − csc2 u
dx
dx
d
du
( sec u ) = sec u tg u
dx
dx
d
du
( csc u ) = − csc u ctg u
dx
dx
d
du
( vers u ) = sen u
dx
dx
DERIV DE FUNCS TRIGO INVER
1
d
du
⋅
( ∠ sin u ) =
dx
1 − u 2 dx
1
d
du
⋅
( ∠ cos u ) = −
dx
1 − u 2 dx
1
d
du
⋅
( ∠ tg u ) =
dx
1 + u 2 dx
1
d
du
⋅
( ∠ ctg u ) = −
dx
1 + u 2 dx
1
d
du ⎧+ si u > 1
⋅ ⎨
( ∠ sec u ) = ±
dx
u u 2 − 1 dx ⎩− si u < −1
1
d
du ⎧− si u > 1
⋅ ⎨
( ∠ csc u ) = ∓
dx
u u 2 − 1 dx ⎩+ si u < −1
1
d
du
⋅
( ∠ vers u ) =
dx
2u − u 2 dx
DERIVADA DE FUNCS HIPERBÓLICAS
d
du
sinh u = cosh u
dx
dx
d
du
cosh u = sinh u
dx
dx
d
du
tgh u = sech 2 u
dx
dx
d
du
ctgh u = − csch 2 u
dx
dx
d
du
sech u = − sech u tgh u
dx
dx
d
du
csch u = − csch u ctgh u
dx
dx
DERIVADA DE FUNCS HIP INV
d
du
1
senh −1 u =
⋅
dx
1 + u 2 dx
d
du
±1
cosh −1 u =
⋅ , u >1
dx
u 2 − 1 dx
d
1 du
⋅ , u <1
tgh −1 u =
dx
1 − u 2 dx
d
1 du
−1
⋅ , u >1
ctgh u =
dx
1 − u 2 dx
d
du ⎧−
∓1
⎪ si
sech −1 u =
⋅ ⎨
dx
u 1 − u 2 dx ⎪⎩ + si
-1
⎪⎧+ si cosh u > 0
⎨
-1
⎪⎩− si cosh u < 0
sech −1 u > 0, u ∈ 0,1
sech −1 u < 0, u ∈ 0,1
∫ { 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 = 0
m ⋅ ( b − a ) ≤ ∫ f ( x ) dx ≤ M ⋅ ( b − a )
b
b
a
a
b
b
a
a
b
c
b
a
a
c
b
a
a
b
a
a
b
a
⇔ m ≤ f ( x ) ≤ M ∀x ∈ [ a, b ] , m, M ∈
∫ f ( x ) dx ≤ ∫ g ( x ) dx
b
b
a
a
⇔ f ( x ) ≤ g ( x ) ∀x ∈ [ a , b ]
∫ f ( x ) dx ≤ ∫ f ( x ) dx si a < b
b
b
a
a
INTEGRALES
∫ adx =ax
∫ af ( x ) dx = a ∫ f ( x ) dx
∫ ( u ± v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ±
∫ udv = uv − ∫ vdu ( Integración por partes )
u n+1
∫ u du = n + 1
n
du
∫u
= ln u
u
u
au ⎧a > 0
u
∫ a du = ln a ⎨⎩a ≠ 1
au ⎛
−1
1 ⎞
∫ ua du = ln a ⋅ ⎜⎝ u − ln a ⎟⎠
u
∫ ue du = e ( u − 1)
∫ ln udu =u ln u − u = u ( ln u − 1)
u
u
1
u
( u ln u − u ) = ( ln u − 1)
ln a
ln a
u2
∫ u log a udu = 4 ⋅ ( 2log a u − 1)
u2
∫ u ln udu = 4 ( 2ln u − 1)
INTEGRALES DE FUNCS TRIGO
∫ log
a
udu =
∫ sin udu = − cos u
∫ cos udu = sin u
∫ sec udu = tg u
∫ csc udu = − ctg u
∫ sec u tg udu = sec u
∫ csc u ctg udu = − csc u
∫ tg udu = − ln cos u = ln sec u
∫ ctg udu = ln sin u
∫ sec udu = ln sec u + tg u
∫ csc udu = ln csc u − ctg u
1
= ln tgh u
2
INTEGRALES DE FRAC
du
1
u
∫ u 2 + a 2 = a ∠ tg a
u
1
= − ∠ ctg
a
a
du
1 u−a
2
2
∫ u 2 − a 2 = 2a ln u + a ( u > a )
du
1 a+u
2
2
∫ a 2 − u 2 = 2a ln a − u ( u < a )
INTEGRALES CON
∫
du
= ∠ sin
a2 − u2
∫
∫ ctg
2
2
(
du
= ln u + u 2 ± a 2
u 2 ± a2
(
∫e
udu = − ( ctg u + u )
∫ u sin udu = sin u − u cos u
∫e
∫ u cos udu = cos u + u sin u
au
au
sin bu du =
cos bu du =
INTEGRALES DE FUNCS TRIGO INV
2
2
a 2 + b2
e au ( a cos bu + b sin bu )
a2 + b2
1
1
∫ sec u du = 2 sec u tg u + 2 ln sec u + tg u
ALGUNAS SERIES
+
+
2
2
−1
= u∠ sec u − ∠ cosh u
∫ ∠ csc udu = u∠ csc u + ln ( u +
u2 − 1
)
)
= u∠ csc u + ∠ cosh u
INTEGRALES DE FUNCS HIP
2
e au ( a sin bu − b cos bu )
f '' ( x0 )( x − x0 )
f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) +
2
2
)
3
∫ ∠ sin udu = u∠ sin u + 1 − u
∫ ∠ cos udu = u∠ cos u − 1 − u
∫ ∠ tg udu = u∠ tg u − ln 1 + u
∫ ∠ ctg udu = u∠ ctg u + ln 1 + u
∫ ∠ sec udu = u∠ sec u − ln ( u + u
∫ sinh udu = cosh u
∫ cosh udu = sinh u
∫ sech udu = tgh u
∫ csch udu = − ctgh u
∫ sech u tgh udu = − sech u
∫ csch u ctgh udu = − csch u
)
1
u
∫ u a 2 ± u 2 = a ln a + a 2 ± u 2
1
du
a
∫ u u 2 − a 2 = a ∠ cos u
1
u
= ∠ sec
a
a
u 2
a2
u
2
2
2
∫ a − u du = 2 a − u + 2 ∠ sen a
2
u 2
a
2
2
2
2
2
∫ u ± a du = 2 u ± a ± 2 ln u + u ± a
MÁS INTEGRALES
udu =
n ≠ −1
u
a
du
u 1
− sin 2u
2 4
u 1
2
∫ cos udu = 2 + 4 sin 2u
2
∫ tg udu = tg u − u
∫ sin
u
a
= −∠ cos
2
INTEGRALES DEFINIDAS, PROPIEDADES
Nota. Para todas las fórmulas de integración deberá
agregarse una constante arbitraria c (constante de
integración).
b
∫ e du = e
Jesús Rubí M.
∫ tgh udu = ln cosh u
∫ ctgh udu = ln sinh u
∫ sech udu = ∠ tg ( sinh u )
∫ csch udu = − ctgh ( cosh u )
2
d
du
1
csch −1 u = −
⋅ , u≠0
dx
u 1 + u 2 dx
a
INTEGRALES DE FUNCS LOG & EXP
f ( n ) ( x0 )( x − x0 )
n!
f ( x ) = f (0) + f ' ( 0) x +
+
+
f ( n) ( 0 ) x n
2!
n
: Taylor
f '' ( 0 ) x 2
2!
: Maclaurin
n!
x 2 x3
xn
+ + +
+
n!
2! 3!
x 3 x5 x 7
x 2 n −1
n −1
−
+ + ( −1)
sin x = x − +
3! 5! 7!
( 2n − 1)!
ex = 1 + x +
cos x = 1 −
x2 x4 x6
+
+
−
2! 4! 6!
+ ( −1)
n −1
x 2 n− 2
( 2n − 2 ) !
n
x2 x3 x 4
n −1 x
+ −
+ + ( −1)
n
2
3
4
2 n −1
x3 x 5 x 7
n −1 x
∠ tg x = x − + −
+ + ( −1)
3
5
7
2n − 1
ln (1 + x ) = x −
2
Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3)
ALFABETO GRIEGO
Mayúscula Minúscula Nombre
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Α
Β
Γ
∆
Ε
Ζ
Η
Θ
Ι
Κ
Λ
Μ
Ν
Ξ
Ο
Π
Ρ
Σ
Τ
Υ
Φ
Χ
Ψ
Ω
α
β
γ
δ
ε
ζ
η
θ
ϑ
ι
κ
λ
µ
ν
ξ
ο
π ϖ
ρ
ς
τ
υ
φ ϕ
χ
ψ
ω
σ
Alfa
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Teta
Iota
Kappa
Lambda
Mu
Nu
Xi
Omicron
Pi
Rho
Sigma
Tau
Ipsilon
Phi
Ji
Psi
Omega
Equivalente
Romano
A
B
G
D
E
Z
H
Q
I
K
L
M
N
X
O
P
R
S
T
U
F
C
Y
W
NOTACIÓN
sin
cos
tg
Seno.
Coseno.
Tangente.
sec
csc
ctg
Secante.
Cosecante.
Cotangente.
vers Verso seno.
arcsin θ =
sin 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.
= {…, −2, −1,0,1, 2,…}
Conjunto de enteros.
Conjunto de números racionales.
c
Conjunto de números irracionales.
= {1, 2,3,…} Conjunto de números naturales.
Conjunto de números complejos.
http://www.geocities.com/calculusjrm/
Jesús Rubí M.