Formulario de Cálculo Diferencial e Integral (Página 1 de 2)

Formulario de Cálculo Diferencial e Integral VER.4.9 Jesús Rubí Miranda (jesusrubim@yahoo.com) http://www.geocities.com/calculusjrm/ VALOR ABSOLUTO ⎧a si a ≥ 0 a =⎨ ⎩− a si a < 0 a = −a

ab = a b ó

∏a

a+b ≤ a + b ó

n

= ∏ ak

k

∑ ca

k =1

n

∑a k =1

a ⋅a = a

k =1

k

∑(a

≤ ∑ ak

k =1

k =1

p+q

k

∑(a k =1

0 30 45 60 90

sen

ctg

sec

csc

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

y = ∠ sen x y = ∠ cos x

⎡ π π⎤ y ∈ ⎢− , ⎥ ⎣ 2 2⎦ y ∈ [ 0, π ] y∈ −

1 y = ∠ ctg x = ∠ tg x

k =1

tg 0

cos

0 12

y = ∠ tg x

+ an = ∑ ak

k =1

n

n

k =1

k =1

+ bk ) = ∑ ak + ∑ bk

k

− ak −1 ) = an − a0

, 2 2

= a pq

(a ⋅ b)

p

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

= ap ⋅bp

n

∑ ar

a ⎛a⎞ ⎜ ⎟ = p b ⎝b⎠

p

k −1

k =1

a p/q = a p q

LOGARITMOS log a N = x ⇒ a = N x

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

1+ 3 + 5 +

log b N ln N log a N = = log b a ln a

-0.5

cos (θ + π ) = − cosθ

2

( a − b ) ⋅ ( a + ab + b ) = a − b ( a − b ) ⋅ ( a3 + a 2 b + ab2 + b3 ) = a 4 − b4 ( a − b ) ⋅ ( a 4 + a3b + a 2 b2 + ab3 + b 4 ) = a5 − b5 2

n

2

3

3

( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n ⎝ k =1

-4

-2

0

2

4

6

n

8

2.5

0.5 0 -0.5 -1 -1.5

+ xk )

csc x sec x ctg x

-2 -2.5 -8

-6

-4

-2

0

2

4

6

8

Gráfica 3. Las funciones trigonométricas inversas arcsen x , arccos x , arctg x :

n! =∑ x1n1 ⋅ x2n2 n1 !n2 ! nk !

nk k

x

HIP θ CA

cos (α ± β ) = cos α cos β ∓ sen α sen β

1

tg α ± tg β tg (α ± β ) = 1 ∓ tg α tg β sen 2θ = 2sen θ cosθ

-2 -3

CO

π⎞ ⎛ sen θ = cos ⎜θ − ⎟ 2⎠ ⎝

sen (α ± β ) = sen α cos β ± cos α sen β

-1

∀n ∈

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

2

0

arc sen x arc cos x arc tg x -2

-1

0

1

2

cos 2θ = cos 2 θ − sen 2 θ 3

e x − e− x 2 e x + e− x cosh x = 2 senh x e x − e − x tgh x = = cosh x e x + e − x 1 e x + e− x ctgh x = = tgh x e x − e − x 1 2 sech x = = cosh x e x + e − x 1 2 csch x = = senh x e x − e − x senh x =

senh : n

π⎞ ⎛ cosθ = sen ⎜θ + ⎟ 2⎠ ⎝

4

3

CONSTANTES π = 3.14159265359… e = 2.71828182846… TRIGONOMETRÍA CO 1 sen θ = cscθ = HIP sen θ CA 1 cosθ = secθ = HIP cosθ sen θ CO 1 tg θ = = ctg θ = cosθ CA tg θ

tg (θ + nπ ) = tg θ

sen ( nπ ) = 0 tg ( nπ ) = 0

1

+ ( 2n − 1) = n 2

n

cos (θ + nπ ) = ( −1) cos θ

cos ( nπ ) = ( −1)

2 1.5

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

( x1 + x2 +

tg (θ + π ) = tg θ sen (θ + nπ ) = ( −1) sen θ

Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x :

k =1

( a + b) ⋅ ( a − b) = a − b 2 ( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b2 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 + 3ab2 + b3 3 ( a − b ) = a3 − 3a 2b + 3ab2 − b3 2 ( a + b + c ) = a 2 + b2 + c 2 + 2ab + 2ac + 2bc 2

tg ( −θ ) = − tg θ

tg (θ + 2π ) = tg θ

sen x cos x tg x

sen α ⋅ cos β =

cos ( −θ ) = cosθ

sen (θ + π ) = − sen θ

-6

5

tg α + tg β ctg α + ctg β FUNCIONES HIPERBÓLICAS

n

n

ALGUNOS PRODUCTOS a ⋅ ( c + d ) = ac + ad

tg α ⋅ tg β =

0

-2 -8

n! = ∏ k

log10 N = log N y log e N = ln N

sen ( −θ ) = − sen θ

1 + ctg 2 θ = csc2 θ

cos (θ + 2π ) = cosθ

-1.5

1 2 ( n + n) 2 k =1 n 1 k 2 = ( 2n3 + 3n 2 + n ) ∑ 6 k =1 n 1 4 3 k = ( n + 2n3 + n 2 ) ∑ 4 k =1 n 1 k 4 = ( 6n5 + 15n 4 + 10n3 − n ) ∑ 30 k =1

sen (α ± β ) cos α ⋅ cos β

tg 2 θ + 1 = sec2 θ

sen θ + cos2 θ = 1

sen (θ + 2π ) = sen θ

n

1 1 = 2sen (α + β ) ⋅ cos (α − β ) 2 2 1 1 = 2 sen (α − β ) ⋅ cos (α + β ) 2 2 1 1 = 2 cos (α + β ) ⋅ cos (α − β ) 2 2 1 1 = −2sen (α + β ) ⋅ sen (α − β ) 2 2

1 ⎡sen (α − β ) + sen (α + β ) ⎦⎤ 2⎣ 1 sen α ⋅ sen β = ⎣⎡cos (α − β ) − cos (α + β ) ⎦⎤ 2 1 cos α ⋅ cos β = ⎣⎡cos (α − β ) + cos (α + β ) ⎦⎤ 2

0

2

-1

∑k =

tg α ± tg β = arc ctg x arc sec x arc csc x

y ∈ 0, π

0.5

=

cos α − cos β

-2 -5

1

k =1

cos α + cos β 2

-1

2

n

3

π π

1.5

n

sen α + sen β sen α − sen β

4

0

1 y ∈ [ 0, π ] x 1 ⎡ π π⎤ y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥ x ⎣ 2 2⎦ Gráfica 1. Las funciones trigonométricas: sen x , cos x , tg x :

= c ∑ ak

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

1

y = ∠ sec x = ∠ cos

k

n

θ

n

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

(a )

par

n

n

n

ap = a p−q aq

p

k +1

k =1

n

EXPONENTES q

⎞ a n − k b k −1 ⎟ = a n − b n ∀ n ∈ ⎠ SUMAS Y PRODUCTOS

( a + b ) ⋅ ⎜ ∑ ( −1) ⎝ k =1

impar

n

n

k =1

n

∑ c = nc

a ≥0y a =0 ⇔ a=0

p q

⎛ n ⎞ k +1 ( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈ ⎝ k =1 ⎠

a1 + a2 +

a ≤ a y −a≤ a

p

Jesús Rubí M.

( a + b ) ⋅ ( a 2 − ab + b2 ) = a3 + b3 ( a + b ) ⋅ ( a3 − a 2 b + ab2 − b3 ) = a 4 − b4 ( a + b ) ⋅ ( a 4 − a3b + a 2 b2 − ab3 + b4 ) = a5 + b5 ( a + b ) ⋅ ( a5 − a 4 b + a3b2 − a 2 b3 + ab4 − b5 ) = a 6 − b6

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

cosh : tgh : ctgh :

→ → [1, ∞ → −1,1 − {0} → −∞ , −1 ∪ 1, ∞

sech :

→ 0,1]

csch :

− {0} →

− {0}

Gráfica 5. Las funciones hiperbólicas senh x ,

cosh x , tgh x : 5 4 3 2 1 0 -1 -2 se nh x co sh x tgh x

-3 -4 -5

0

5

FUNCIONES HIPERBÓLICAS INV

( (

) )

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

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

cosh 2 x − senh 2 x = 1 1 − tgh 2 x = sech 2 x ctgh x − 1 = csch x 2

senh ( − x ) = − senh x cosh ( − x ) = cosh x tgh ( − x ) = − tgh x

senh ( x ± y ) = senh x cosh y ± cosh x senh y cosh ( x ± y ) = cosh x cosh y ± senh x senh y tgh x ± tgh y 1 ± tgh x tgh y senh 2 x = 2senh x cosh x tgh ( x ± y ) =

cosh 2 x = cosh x + senh x 2

tgh 2 x =

2

2 tgh x 1 + tgh 2 x

1 ( cosh 2 x − 1) 2 1 2 cosh x = ( cosh 2 x + 1) 2 cosh 2 x − 1 tgh 2 x = cosh 2 x + 1

senh 2 x =

tgh x =

senh 2 x cosh 2 x + 1

e x = cosh x + senh x e − x = cosh x − senh x OTRAS

ax + bx + c = 0 2

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

⇒ x=

exp (α ± iβ ) = eα ( cos β ± i sen 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 =0 lim x →0 x ex −1 =1 lim x →0 x x −1 =1 lim 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 ) = ncxn−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 du u ) = nu n −1 ( 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 ( sen u ) = cos u dx dx d du ( cos u ) = − sen 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 d du 1 ⋅ ( ∠ sen u ) = dx 1 − u 2 dx

d du 1 ⋅ ( ∠ cos u ) = − dx 1 − u 2 dx d 1 du ⋅ ( ∠ tg u ) = dx 1 + u 2 dx d 1 du ( ∠ ctg u ) = − 2 ⋅ dx 1 + u dx d du ⎧ + si u > 1 1 ( ∠ sec u ) = ± 2 ⋅ ⎨ dx u u − 1 dx ⎩ − si u < −1 d du ⎧− si u > 1 1 ⋅ ⎨ ( ∠ csc u ) = ∓ dx u u 2 − 1 dx ⎩+ si u < −1 d du 1 ⋅ ( ∠ vers u ) = dx 2u − u 2 dx

Jesús Rubí M. DERIVADA DE FUNCS HIPERBÓLICAS d du senh u = cosh u dx dx d du cosh u = senh 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 1 du senh −1 u = ⋅ dx 1 + u 2 dx -1 d ±1 du ⎪⎧+ si cosh u > 0 cosh −1 u = ⋅ , u >1 ⎨ -1 dx u 2 − 1 dx ⎪⎩− si cosh u < 0 d 1 du ⋅ , u <1 tgh −1 u = dx 1 − u 2 dx d du 1 ⋅ , u >1 ctgh −1 u = dx 1 − u 2 dx −1 ∓1 d du ⎪⎧− si sech u > 0, u ∈ 0,1 ⋅ ⎨ sech −1 u = −1 dx u 1 − u 2 dx ⎪⎩ + si sech u < 0, u ∈ 0,1 d 1 du −1 csch u = − ⋅ , u≠0 dx u 1 + u 2 dx

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

∫ { 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

b

a

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 n ∫ u du = n + 1 n ≠ −1 du ∫ u = ln u

INTEGRALES DE FUNCS LOG & EXP

∫ e du = e u

u

∫ ctgh udu = ln senh u ∫ sech udu = ∠ tg ( senh u ) ∫ csch udu = − ctgh ( cosh u )

a u ⎧a > 0

∫ a du = ln a ⎨⎩a ≠ 1 u

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 2 u ∫ u log a udu = 4 ⋅ ( 2log a u − 1) u2 ∫ u ln udu = 4 ( 2ln u − 1) INTEGRALES DE FUNCS TRIGO

∫ log

a

∫ tgh udu = ln cosh u

udu =

∫ sen udu = − cos u ∫ cos udu = sen 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 sen 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 1 u = − ∠ ctg a a 1 u−a du 2 2 = ln ∫ u 2 − a 2 2a u + a ( u > a ) 1 a+u du 2 2 ∫ a 2 − u 2 = 2a ln a − u ( u < a ) INTEGRALES CON

2

du a2 − u2

= ∠ sen

= −∠ cos

2

2

udu =

du u 2 ± a2

(

= ln u + u 2 ± a 2

2

2

∫ u sen udu = sen u − u cos u ∫ u cos udu = cos u + u sen u INTEGRALES DE FUNCS TRIGO INV

∫ ∠ sen udu = u∠ sen 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 2

du

a2 ± u2 du

2

+

+

f(

n)

n!

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

( x0 )( x − x0 )

f ( x ) = f ( 0) + f '( 0) x +

2!

n

: Taylor

f '' ( 0 ) x 2 2!

( 0 ) x n : Maclaurin n! x 2 x3 xn ex = 1 + x + + + + + 2! 3! n! x3 x 5 x 7 x 2 n −1 n −1 sen x = x − + − + + ( −1) 3! 5! 7! ( 2n − 1)! +

cos x = 1 −

+

f(

n)

x2 x4 x6 + − + 2! 4! 6!

+ ( −1)

n −1

x 2n−2

( 2n − 2 )!

n x 2 x3 x 4 n −1 x + − + + ( −1) 2 3 4 n 2 n −1 x3 x5 x7 n −1 x ∠ tg x = x − + − + + ( −1) 3 5 7 2n − 1

ln (1 + x ) = x −

)

f '' ( x0 )( x − x0 )

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

2

2

)

(

∫ tg udu = tg u − u ∫ ctg udu = − ( ctg u + u )

∫ senh udu = cosh u ∫ cosh udu = senh u ∫ sech udu = tgh u ∫ csch udu = − ctgh u ∫ sech u tgh udu = − sech u ∫ csch u ctgh udu = − csch u

u a

1 u = ln a a + a2 ± u 2 1 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 e au ( a sen bu − b cos bu ) au e sen bu du = ∫ a2 + b2 eau ( a cos bu + b sen bu ) au ∫ e cos bu du = a 2 + b2 1 1 3 ∫ sec u du = 2 sec u tg u + 2 ln sec u + tg u ALGUNAS SERIES

∫u

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

∫ sen

u a

2

Formulario de calculo