Formulas de Calculo Todas Derivadas Integrales Limites

Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3)

http://www.geocities.com/calculusjrm/

Jesús Rubí M.

Fórmulas de

Cálculo Diferencial

e Integral

V

ER

.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 = = = = ≥ ⎧ = ⎨_{− <} ⎩ = − ≤ − ≤ ≥ = ⇔ = = = + ≤ + ≤

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EXPONENTES

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/ p q p q p p q q q p pq p p p p p p q p 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 +

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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 = − ⋅ + + = − − ⋅ + + + = − − ⋅ + + + + = − ⎛ ⎞ − ⋅⎜_⎝

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⎟_⎠= − ∀ ∈

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2 2 3 3 3 2 2 3 4 4 4 3 22 3 4 5 5

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2 2 3 3 3 2 2 3 4 4 4 3 22 3 4 5 5 5 4 32 23 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 + ⋅ − + = + + ⋅ − + − = − + ⋅ − + − + = + + ⋅ − + − + − = −

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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 + − − = + − − = ⎛ ⎞ + ⋅⎜_⎝ − ⎟_⎠= + ∀ ∈ ⎛ ⎞ + ⋅⎜ − ⎟= − ∀ ∈ ⎝ ⎠

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SUMAS Y PRODUCTOS n

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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 = = = = = = = − = + + + = = = + = + − = −

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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 n n 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 = − = = = = = = + − = + − ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ + − − = = − − = + = + + = + + = + + − + + + + − = = ⎛ ⎞₌ ⎜ ⎟ ⎝ ⎠

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0 ! , ! ! n n n k k k k n n k k n x y x y k − = ≤ − ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠

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1 2 1 2 1 2 1 2 ! ! ! ! k n n n n k k k n x x x x x x n n n + + + =

_∑

⋅ CONSTANTES 9… 3.1415926535 2.71828182846 e 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=180 π 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

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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 + = + = + =

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sin sin cos cos tg tg θ θ θ θ θ θ − = − − = − = −

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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 + = + = + = + = − + = − + = + = − + = − + =

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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 ⎛ ⎞ = _⎜ − _⎟ ⎝ ⎠ ⎛ ⎞ = ⎜_⎝ + ⎟_⎠

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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

cos 2 cos sin

2 tg tg 2 1 tg 1 sin 1 cos 2 2 1 cos 1 cos 2 2 1 cos 2 tg 1 cos 2 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 ± = ± ± = ± ± = = = − = − = − = + − = + ∓ ∓

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1 1

sin sin 2 sin cos

2 2

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sin sin 2 sin cos

2 2

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cos cos 2 cos cos

2 2

1 1

cos cos 2 sin 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 + = + ⋅ − − = − ⋅ + + = + ⋅ − − = − + ⋅ −

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sin tg tg cos cos α β α β α β ± ± = ⋅

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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

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sinh : cosh : 1, tgh : 1,1 ctgh : 0 , 1 1, sech : 0 ,1 csch : 0 0 → → ∞ → − − → −∞ − ∪ ∞ → − → −

Gráfica 5. Las funciones hiperbólicas sinhx,

coshx, tgh x : -5 0 5 -4 -3 -2 -1 0 1 2 3 4 5 senh x cosh x tgh x

FUNCIONES HIPERBÓLICAS INV

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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_{x sinh}2_{x 1}

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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 − = − = − = − − = − = − − =

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2 2 2

sinh sinh cosh cosh sinh

cosh cosh cosh sinh sinh

tgh tgh

tgh

1 tgh tgh

sinh 2 2 sinh cosh

cosh 2 cosh sinh

2 tgh 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 ± = ± ± = ± ± ± = ± = = + = +

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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

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2 2 2 0 4 2 4 discriminante exp cos sin si ,

ax bx c b b ac x a b ac i eα i α β β β α β + + = − ± − ⇒ = − = ± = ± ∈ LÍMITES

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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

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0 0 1 lim lim 0 x _{x x} n n f x x f x df 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 ∆ → ∆ → − + ∆ − ∆ = = = ∆ ∆ = = = ± ± ± = ± ± ± =

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2 1 n 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 dx d u dx v v d du u nu dx dx − = + = + + − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ =

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1 2 1 2 (Regla de la Cadena) 1 donde dF dF du dx du dx du dx dx du dF du dF dx dx du x f t f t dy dt dy dx dx dt f t y f t = ⋅ = = = ⎧ ′ ⎪ = = _′ ⎨ ₌ ⎪⎩

DERIVADA DE FUNCS LOG & EXP

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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 e d 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 − = = ⋅ = ⋅ = ⋅ > = ⋅ = ⋅ = + ⋅ ⋅ 1 d du dx du a≠

DERIVADA DE FUNCIONES TRIGO

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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

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2 2 2 2 2 2 2 sin 1 1 cos 1 1 tg 1 1 ctg 1 si 1 1 sec si 1 1 si 1 1 csc si 1 1 1 vers 2 u dx u dx d du u dx u dx d du u dx u dx d du u dx u dx u d du u u dx u u dx u d du u u dx u u dx d du u dx u u dx ∠ = ⋅ − ∠ = − ⋅ − ∠ = ⋅ + ∠ = − ⋅ + + > ⎧ ∠ = ± ⋅ ⎨ 1 d 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 -1 2 1 2 1 2 1 1 1 2 senh 1 si cosh 0 1 cosh , 1 si cosh 0 1 1 tgh , 1 1 1 ctgh , 1 1 si sech 0, 0,1 1 sech si sech 0, 0,1 1 u dx u dx u d du u u dx u dx u d du u u dx u dx d du u u dx u dx u u d du u dx u u dx u u − − − − − − − = ⋅ + ⎧+ > ± ⎪ = ₋ ⋅ > ⎨_{− <} ⎪⎩ = ⋅ < − = ⋅ > − ⎧− > ∈ ⎪ = ⋅ ⎨ + < ∈ − _⎩ ∓ 1 d du 1 2 1 csch , 0 1 d du u u dx u u dx − ⎪ = − ⋅ ≠ +

INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá

agregarse una constante arbitraria c (constante de

integración).

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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 ± = ± = ⋅ ∈ = + = − = ⋅ − ≤ ≤ ⋅ − ⇔ ≤ ≤ ∀ ∈ ∈ ≤ ⇔ ≤ ∀ ∈ ≤ <

∫

∫

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∫

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INTEGRALES

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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

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2 2 0 1 ln 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 a a a du a a 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 = − = = = + = −

∫

∫

∫

∫

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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

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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 = = = = − = − = −

∫

∫

∫

∫

∫

∫

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1 tgh ln cosh 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

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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

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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 a a u u a du u u a u a du u a u a u a a u du a a u u 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

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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 u du u u u u − = + + = + = + +

∫

∫

∫

au ALGUNAS SERIES

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₎

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2 0 0 0 0 0 0 0 2 2 3 3 5 7 21 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! n n 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 − − − = + − + − + + = + + + + = + + + + + + = − + − + + − − = − + −

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2 2 1 2 3 4 1 3 5 7 21 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)

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Jesús Rubí M. ALFABETO GRIEGO

ayúscula Minúscula Nombre

M 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.

arcsinθ= sinθ Arco seno de un ángulo θ .

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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.