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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
⎛ 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 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.
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Jesús Rubí M.
