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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 si a ≥ 0 a =⎨ ⎩ −a si a < 0 a = −a

∏a

= ∏ ak

k

k=1

n

∑a

k

k =1

n

≤ ∑ ak k =1

EXPONENTES p+ q

a ⋅a = a a a

2 3

4

(a + b )⋅ ⎛⎜ ∑(− 1) k 1 a n− kb k−1 ⎞⎟ = a n + b n n

+

⎝ k =1

⎠

5

6

6

∀ n ∈  impar

⎛ n ⎞ +1 (a + b )⋅ ⎜ ∑(− 1) k a n− kb k−1 ⎟ = a n − b n ∀ n ∈  par ⎝ k =1 ⎠ SUMAS Y PRODUCTOS

∑ca

n

= c ∑ak

k

k =1

k =1

n

n

n

k =1

k =1

∑( a

+ bk ) = ∑ ak + ∑bk

∑( a

− ak −1) = an − a 0

k

k =1 n

k

sin 0 12

1

tg

cos

ctg

sec

csc

0 1 ∞ ∞ 1 3 2 3 2 3 2 1 3 2 1 2 2 2 1 1 3 1 3 3 2 12 2 2 3 0 0 1 ∞ ∞ 1

⎡ π π ⎤ y ∈ ⎢− , ⎥ ⎣ 2 2⎦ y = ∠ cos x y ∈ [0, π ]

=a

q

(a )

q

( a ⋅ b)

=a p

n

y = ∠ tg x

y∈ −

2 1 x y = ∠ sec x = ∠ cos 1 x 1 y = ∠ csc x = ∠ sen x

,

2

p

n

p

q

= a

∑ar

k −1

k =1 n

p

LOGARITMOS log a N = x ⇒ a x = N

log a M = log a M − log a N N r log a N = r log a N log b N lnN = log a N = logb a ln a log 10 N = log N y log e N = ln N ALGUNOS PRODUCTOS a ⋅ (c + d ) = ac + ad

( a + b) ⋅( a − b) = a2 − b2 2 ( a + b) ⋅( a + b) = ( a + b) = a2 + 2ab + b2 ( a − b) ⋅( a − b) = ( a − b) 2 = a2 − 2ab + b2 ( x + b) ⋅( x + d ) = x2 + ( b + d) x + bd ( ax + b) ⋅( cx + d) = acx2 +( ad + bc) x + bd ( a + b) ⋅( c + d) = ac + ad + bc + bd ( a + b) 3 = a 3 + 3 a 2b + 3 ab 2 + b 3 ( a − b) = a3 − 3a2 b + 3ab2 − b3 2 ( a + b + c) = a2 + b2 + c2 + 2ab + 2ac + 2bc 3

( a − b) ⋅ ( a 2 + ab + b2 ) = a 3 − b 3 ( a − b) ⋅ ( a3 + a2 b + ab 2 + b 3 ) = a 4 − b 4 ( a − b) ⋅ ( a 4 + a 3b + a 2b 2 + ab 3 + b 4 ) = a 5 − b 5 n

⎝ k =1

⎠

− bn ∀ n ∈ 

2

k =1 n

∑k

2

y ∈ [ 0, π ]

sin θ + cos 2θ = 1 2 2 1+ ctg θ = csc θ

⎡ π π⎤ y ∈ ⎢− , ⎥ ⎣ 2 2⎦

sin (− θ )= − sinθ

2

2 2 tg θ + 1 = sec θ

∑k

3

1 n4 n3 n ( +2 + 4

6

3

sin (θ +π

)

2

-4

-2

0

2

4

6

sin (θ +nπ 8

2 .5

2 1+ 3+ 5+  + ( 2n − 1) = n

0 .5 0

n

-0.5 -1

k =1

-1.5

⎛n ⎞ n! , k≤ n ⎜ ⎟= ⎝k ⎠ (n −k ) !k !

(x + y )

n

(x1 + x 2

csc x -2

se c x ctg x

-2.5 -8

⎛n ⎞ = ∑ ⎜ ⎟ xn −k yk k=0 ⎝k ⎠

-6

-4

-2

0

2

4

6

8

n

+  + xk

)

n

Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctgx :

n! x n 1 ⋅ x2n 2  xkn k =∑ n1 !n 2 !nk ! 1

CA

a rc se n x arc co s x a rc tg x

CO

⎛ 2n + 1 ⎞ = − n π ⎟ ( 1) sin ⎜ ⎝ 2 ⎠ ⎛ 2n +1 ⎞ = π⎟ 0 cos⎜ 2 ⎝ ⎠ n+ tg ⎛⎜ 2 1π ⎞⎟ = ∞ ⎝ 2 ⎠ ⎛ π⎞ sin θ = cos⎜ θ − ⎟ 2⎠ ⎝ π⎞ ⎛ cosθ = sin ⎜ θ + ⎟ 2⎠ ⎝

tg α± tg β 1 ∓ tg α tg β sin 2θ = 2 sinθ cosθ

-1

HIP

n

tg( α ± β) =

0

-3

sin (n π ) = 0

) = sinα cosβ ± cosα sinβ cos (α ± β ) = cos α cos β ∓ sin α sin β

1

-2

n

tg( θ + nπ) = tg θ

sin (α ±β

2

π radianes=180

θ

4

3

CONSTANTES π = 3.14159265359… e = 2.71828182846 … TRIGONOMETRÍA CO 1 senθ = cscθ = senθ HIP CA 1 cosθ = sec θ = cos θ HIP senθ CO 1 tgθ = ctgθ = = cosθ CA tgθ

) =( −1)n sinθ

cos (θ +n π ) = ( −1) cos θ

tg ( nπ ) = 0

1

n != ∏ k

= −sinθ

cos( nπ ) = ( − 1)

2 1 .5

)

tg( θ + π) = tg θ se n x cos x tg x -6

Gráfica 2. Las funciones trigonométricas cscx , sec x , ctgx :

1 ∑=1k = 30 (6n 5 + 15n 4 + 10n 3 − n ) k

= sinθ

cos (θ + π ) = − cos θ

-1

4

)

tg( θ + 2 π) = tg θ

-1.5

+ 3n 2 + n )

=

k =1 n

( 2n

1

tg ( −θ ) = − tg θ cos (θ + 2π ) = cos θ

0

-2 -8

cos (−θ ) = cos θ sin (θ +2π

-0.5

+n)

=

k =1 n

log a MN = log a M + loga N

( a − b) ⋅ ⎛⎜ ∑ an − k bk −1 ⎞⎟ = an

1 ∑k = 2 (n

0

-2

-1

0

1

5

IDENTIDADES TRIGONOMÉTRICAS

y ∈ 0,π

0 .5

n ( a + l) 2 1− r n a − rl =a = 1 −r 1 −r

tg α ± tg β = a rc ctg x a rc se c x a rc csc x

-2 -5

1

=

= ap ⋅ bp

a ⎛ a⎞ ⎜⎝ b ⎟⎠ = b p p/ q

n

2

Jesús Rubí M.

1 1 (α + β )⋅ cos 2 (α − β ) 2 1 1 sin α − sin β = 2sin (α − β )⋅ cos ( α + β ) 2 2 1 1 cosα + cos β = 2 cos ( α+ β )⋅ cos ( α− β ) 2 2 1 1 cosα − cos β = − 2sin ( α+ β) ⋅ sin ( α− β) 2 2 sin α + sin β = 2sin

0

2

k =1

3

-1

π π

Gráfica 1. Las funciones trigonométricas: sinx , cos x , tg x :

∑⎣⎡a + (k − 1)d⎦⎤ = 2⎣⎡ 2a + (n − 1)d ⎦⎤

pq

4

1

1 .5

p−q

Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x :

y = ∠ sin x

y = ∠ ctg x = ∠ tg

∑c = nc k =1 n

θ 0 30  45 60 90 

k =1

p

p

a

3 2

n

n

k=1

q

− a b + a b − a b + ab − b ) = a − b 4

k =1

n

p

(a + b ) ⋅ ( a

5

n

a ≥0 y a =0 ⇔ a =0

a+b ≤ a + b ó

(a + b )⋅ ( a 3 − a 2b + ab 2 − b 3 ) = a 4 − b 4 (a + b )⋅ ( a 4 − a3b + a 2b 2 − ab3 + b 4 ) = a 5 + b 5

a1 + a2 + + an = ∑ ak

a≤ a y − a≤ a

ab = a b ó

(a + b )⋅ ( a 2 − ab + b2 ) = a3 + b3

2

3

cos 2θ = cos 2 θ − sin 2 θ 2 tg θ tg 2θ = 1− tg 2 θ sin 2θ = 1 (1 − cos 2θ ) 2 1 2 cos θ = (1 + cos 2θ ) 2 1− cos 2θ 2 tg θ = 1 + cos 2θ

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 cosα ⋅ cos β = ⎣⎡ cos ( α − β ) + cos ( α + β ) ⎦⎤ 2 tg α + tg β tg α ⋅ tg β = ctg α + ctgβ FUNCIONES HIPERBÓLICAS x x e − e− sinh x = 2 x −x e +e cosh x = 2 x x x e − e− sinh tgh x = = cosh x ex + 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 = x −x csch x = sinh x e − e sinh : →  sin α ⋅ cos β =

cosh :  → [1, ∞ tgh :  → −1,1 ctgh : − { 0} → −∞ , −1 ∪ 1, ∞ sech :  → 0,1] csch :  −{ 0} →  −{ 0}

Gráfica 5. Las funciones hiperbólicas sinhx , cosh x , tgh x : 5 4 3 2 1 0 -1 -2 s enh x cosh x tgh x

-3 -4 -5

0

5

FUNCIONES HIPERBÓLICAS INV

( (

1 sinh − x = ln x+

) )

x2 + 1 , ∀ x∈ 

1 2 cosh− x = ln x ± x − 1 , x ≥ 1

1 ⎛ 1+ x ⎞ 1 tgh− x = ln ⎜ ⎟, x < 1 2 ⎝ 1 −x ⎠ 1 ⎛ x +1 ⎞ 1 ctgh − x = ln ⎜ , x >1 2 ⎝ x −1 ⎠⎟ ⎛ ± − x2 ⎞ sech −1 x = ln ⎜ 1 1 ⎟, 0 < x ≤ 1 ⎜ ⎟ x ⎝ ⎠ ⎛1 x2 + 1 ⎞ 1 ⎟, x ≠ 0 csch − x = ln ⎜ + ⎜x x ⎟⎠ ⎝

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Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) IDENTIDADES DE FUNCS HIP 2

2

cosh x − sinh x = 1 2 2 1− tgh x = sech x

ctgh 2 x − 1 = csch 2 x sinh (− x ) = − sinh x cosh( −x ) = cosh x

d n du ( u ) = nu n −1 dx dx

tgh ( −x ) = − tgh x

sinh ( x ± y ) = sinh xcosh y ± cosh xsinh y cosh( x ± y) = cosh x cosh y± sinh x sinh y tgh x ±tgh y tgh (x ± y )= 1± tgh x tgh y sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x 2 tgh x tgh 2 x = 2 1 + tgh x sinh2 x = 1( cosh 2 x − 1) 2 1 2 cosh x = (cosh 2 x +1 ) 2 cosh 2x − 1 2 tgh x = cosh 2x +1 tgh x =

d ( uv) = u dv+ v du dx dx dx d dw dv du ( uvw) = uv dx + uw dx + vw dx dx d ⎛ u⎞ v (du dx ) − u ( dv dx ) = dx ⎝⎜ v ⎠⎟ v2

sinh 2 x cosh 2 x +1

x e = cosh x + sinh x e −x = cosh x − sinh x

OTRAS ax + bx + c = 0 2

− b ± b 2− 4 ac 2a 2 b −4 ac = discriminante ⇒x =

exp (α ± iβ ) = eα (cos β ± isin β ) si α , β ∈  LÍMITES 1 x

lim( 1 + x ) = e = 2.71828...

x →0

x

1⎞ ⎛ lim 1+ ⎟ = e x→∞ ⎜ x⎠ ⎝ senx =1 lim 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 = lim = lim Dxf (x) = ∆x → 0 ∆x dx ∆ x→ 0 ∆x d (c ) = 0 dx d (cx ) = c dx d cx n ) = ncx n−1 ( dx d u v w ( ± ± ± ) = du ± dv ± dw ±  dx dx dx dx d du ( cu) = c dx dx

dF dF du (Regla de la Cadena) = ⋅ dx du dx du 1 = dx dx du dF dF du = dx dx du ⎧⎪ x = f1 (t ) dy dy dt f 2 ′( t ) = = donde ⎨ dx dx dt f1 ′( t ) ⎪⎩ y = f2 ( t) DERIVADA DE FUNCS LOG & EXP d du dx 1 du ( ln u) = = ⋅ dx u u dx d ( log u) =log e ⋅ du dx u dx d log e du ( log a u ) = ua ⋅ dx a > 0, a ≠ 1 dx

d u ( e ) = e u ⋅ du dx dx d u du u a ) = a lna ⋅ ( dx dx d v v− 1 du ( u ) = vu dx + ln u ⋅ uv ⋅ dv 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) = −csc 2u dx dx d du u u u sec sec tg ( )= dx dx d du ( csc u) = −csc u ctg u dx dx d du ( versu ) = sen u dx dx DERIV DE FUNCS TRIGO INVER d 1 du ⋅ ( ∠ sin u) = 2 dx 1 − u dx d du 1 ⋅ ( ∠ cosu) = − 2 dx 1− u dx d du 1 ⋅ ( ∠ tgu ) = 2 dx 1+ u dx d du 1 ( ∠ ctg u) = − 2 ⋅ dx 1+ u dx d du ⎧+ si u > 1 1 ( ∠ sec u) = ± 2 ⋅ dx ⎨ dx ⎩− si u < −1 u u −1 d du ⎧− si u > 1 1 ∠ csc u) = ∓ ⋅ ( ⎨ 2 dx u u −1 dx ⎩+ si u < −1 d ( ∠ vers u) = dx

1 2 2u − u

⋅

du 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 2 ctgh u = − csch u dx dx d du sech u = − sech u tghu dx dx d du csch u = − csch u ctghu dx dx DERIVADA DE FUNCS HIP INV d du 1 1 ⋅ senh− u = 2 dx 1+ u dx ⎧+ ±1 du ⎪ si cosh u > 0 d 1 ⋅ , u >1 ⎨ cosh− u = 2 -1 dx u − 1 dx ⎩⎪− si cosh u < 0 d 1 du 1 ⋅ , u 1 2⋅ dx 1 − u dx −1 d du ⎪⎧− si sech u > 0, u∈ 0,1 1 ∓ 1 ⋅ ⎨ sech − u= −1 2 dx u 1−u dx ⎪⎩+ si sech u < 0,u ∈ 0,1 -1

d

−1

dx

csch u = −

du

1

⋅ , u ≠0 2 u 1 +u dx

INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de integración). b

b

b

∫ { f ( x) ± g ( x) } dx = ∫ f ( x ) dx ± ∫ g ( x ) dx a

a

b

cf ( x ) dx = c⋅

b

c

b

a

f ( x) dx

∫ ∫ ∫ f ( x) dx = ∫ f ( x) dx+ ∫ f ( x) dx a

a

a

c∈ 

b

a

c

∫ f ( x) dx = − ∫ f ( x) dx ∫ f ( x) dx = 0 b

a

a

b

a

a

m ⋅ ( b − a ) ≤ ∫a f ( x) dx ≤ M ⋅ ( b − a) b

⇔ m ≤ f ( x ) ≤ M ∀ x ∈ [ a, b], m, M ∈  b

b

∫ f ( x) dx ≤ ∫ g( x) dx a

a

⇔ f ( x ) ≤ g ( x ) ∀x ∈ [a ,b ] b

b

a

a

∫ f (x )dx ≤∫

f ( x ) dx si a < b

INTEGRALES

∫ adx = ax ∫ af (x )dx = a ∫ f ( x ) dx ∫ (u ±v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ± ∫ udv = uv− ∫ vdu ( Integración por partes) ∫ u du = n

du

∫u

u n +1 n ≠ −1 n +1

= ln u

INTEGRALES DE FUNCS LOG & EXP

∫ e du = e u

u

a

⎧a > 0

u

∫ a du = ln a ⎨⎩a ≠ 1 u

u

a

⎛

−1

1 ⎞

∫ ua du = ln a ⋅ ⎜⎝ u − ln a ⎟⎠ u

1 = ln tgh u 2 INTEGRALES DE FRAC

∫ ue du = e (u − 1 ) ∫ ln udu =u lnu − u = u ( lnu − 1) u

u

u 1 (u ln u − u ) = (ln u − 1 ) ln a ln a 2 u = ⋅ − u udu u log 2log 1 ( ) a ∫ a 4 2 u ∫ u lnudu = 4 ( 2lnu − 1) INTEGRALES DE FUNCS TRIGO ∫ sin udu = − cos u

∫ log

Jesús Rubí M.

∫ tghudu = ln cosh u ∫ ctgh udu = ln sinh u ∫ sechudu = ∠ tg( sinh u) ∫ csch udu = − ctgh ( coshu )

a

udu =

du 1 u = ∠tg +a2 a a u 1 = − ∠ ctg a a du 1 u− a ∫ u 2 − a 2 = 2a ln u + a du 1 a+ u ∫ a 2 − u 2 = 2a ln a − u

∫u

∫

u 1 − sin 2u 2 4 u 1 2 = + sin 2u udu cos ∫ 2 4

∫

du a −u 2

= ∠ sin

2

2

∫

∫u

u a

(

du

= ln u + u 2 ± a 2

2 2 u ±a

2

INTEGRALES DE FUNCS TRIGO INV

∫ ∠ sinudu = u∠ sinu + 1− u ∫ ∠ cosudu = u∠ cosu − 1− u ∫ ∠ tgudu = u∠ tgu − ln 1+ u ∫ ∠ ctg udu = u∠ ctg u + ln 1+ u ∫ ∠ sec udu = u∠ sec u− ln (u + u 2

du 2 2 u −a

=

a 1 ∠ cos a u

=

u 1 ∠sec a a 2

u a u 2 2 a − u + ∠ sen 2 2 a

a − u du = 2

2

(

2

u2 ± a2 du =

f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) +

2

2

+ +

(n )

f

( x 0 )( x − x0 ) n!

2

2

−1

)

= u∠ sec u − ∠ cosh u u −1 2

)

= u∠ csc u + ∠ cosh u INTEGRALES DE FUNCS HIP sinh udu = cosh u

2

)

u a u2 ± a2 ± ln u + u 2 ± a2 2 2 MÁS INTEGRALES au e ( a sin bu − b cos bu ) au e bu du = 2 2 ∫ sin a +b eau ( a cos bu + b sin bu) au ∫ e cos bu du = 2 2 a +b 1 1 3 ∫ sec u du = 2 sec u tg u+ 2 ln sec u+ tg u ALGUNAS SERIES

∫

2

2

< a 2)

u 1 ∫ u a 2 ± u 2 = a ln a + a 2 ± u 2

∫ tg udu = tg u −u ∫ ctg udu = − ( ctg u + u) ∫ u sinudu = sinu − u cosu ∫ u cosudu = cosu + u sinu

∫ ∫ coshudu = sinhu ∫ sech udu = tgh u ∫ csch udu = − ctgh u ∫ sech u tgh udu = − sechu ∫ csch u ctghudu = − cschu

> a 2)

2

du

udu =

∫ ∠ csc udu = u∠ cscu + ln (u +

2

(u

u a

= −∠cos

2

∫ sin

(u

INTEGRALES CON

∫ cosudu = sinu ∫ sec udu = tg u ∫ csc udu = − ctg u ∫ sec u tg udu = sec u ∫ csc u ctg udu = −cscu ∫ tg udu = − ln cos u = ln sec u ∫ ctg udu = ln sin u ∫ secudu = ln secu + tgu ∫ csc udu = ln cscu − ctgu 2

2

f ( x ) = f (0 ) + f ' ( 0) x + + +

(n )

f

2!

n

: Taylor

2! : Maclaurin

n!

2

f ''( x 0)( x −x 0)

f '' ( 0 ) x 2

( 0) x n

n

3

x x x + + + +  n! 2! 3! 5 7 2n 1 n −1 x x x x − +  + ( − 1) sin x = x − + − 3! 5! 7! ( 2 n −1)! x e = 1+ x+

3

n −1 x x x x − + − + +( − 1) 2! 4! 6! ( 2 n −2) ! 2

cos x = 1 −

4

ln ( 1 + x) = x − ∠ tg x = x −

x

x

2

3

3

+

2n 2

6

2

+ x

x

3

5

5

3

−

− x

7

7

x

4

4

)

+ + ( − 1) n −1

+  + (−1 )

n− 1

xn n

2n −1

x 2n − 1

2

Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) ALFABETO GRIEGO M ayúscula Minúscula Nombre Α Β Γ ∆ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω

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 δ ε ζ η θ

ϑ ι κ λ µ ν ξ ο π ϖ ρ σ ς τ υ φ ϕ χ ψ ω

Alfa Beta Gamma Delta Epsilon Zeta Eta Teta Iota Kappa Lambda Mu Nu Xi Omicron Pi Rho Sigma Tau Ipsilon Phi Ji Ps i 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 sec csc ctg

Seno. Coseno. Tangente. 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.

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Jesús Rubí M....