Formulario de Cálculo Diferencial e Integral
Formulario de Cálculo Diferencial e Integral VER.4.3 Jesús Rubí Miranda (
[email protected]) http://mx.geocities.com/estadisticapapers/ http://mx.geocities.com/dicalculus/ VALOR ABSOLUTO ⎧a si a ≥ 0 a =⎨ ⎩− a si a < 0 a = −a
n
∏a
a+b ≤ a + b ó
n
∑ ca
k =1
n
k =1
k =1
n
k
≤ ∑ ak
∑(a
k =1
∑(a
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 ∞ ∞ 0 12
y = ∠ sen x y = ∠ cos x
⎡ π π⎤ y ∈ ⎢− , ⎥ ⎣ 2 2⎦ y ∈ [ 0, π ] y∈ −
k =1
n
n
k =1
k =1
+ bk ) = ∑ ak + ∑ bk
k
− ak −1 ) = an − a0
n
∑ ar
k −1
k =1
, 2 2
LOGARITMOS log a N = x ⇒ a x = N log a MN = log a M + log a N M log a = log a M − log a N N log a N r = r log a N log b N ln N = log b a ln a
1+ 3 + 5 +
-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
⎠
∀n ∈
-2
0
2
4
6
n
8
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
π radianes=180
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
2.5
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
-4
sen α ⋅ cos β =
cos ( −θ ) = cosθ
sen (θ + π ) = − sen θ
-6
5
tg α + tg β ctg α + ctg β FUNCIONES HIPERBÓLICAS
cos (θ + 2π ) = cosθ
Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x :
n
ALGUNOS PRODUCTOS a ⋅ ( c + d ) = ac + ad
tg α ⋅ tg β =
n
n! = ∏ k
log10 N = log N y log e N = ln N
sen ( −θ ) = − sen θ
1 + ctg 2 θ = csc2 θ
sen (θ + 2π ) = sen θ
sen x cos x tg x
sen (α ± β ) cos α ⋅ cos β
tg 2 θ + 1 = sec2 θ
sen θ + cos2 θ = 1
0
-2 -8
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
-1.5
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
tg α ± tg β = arc ctg x arc sec x arc csc x
IDENTIDADES TRIGONOMÉTRICAS
y ∈ 0, π
-1
∑k =
cos α − cos β
-2 -5
n
p
cos α + cos β 2
-1
1
n (a + l ) 2 1 − r n a − rl =a = 1− r 1− r
3
π π
0.5
=
sen α + sen β sen α − sen β
4
0
2
n
Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x :
1
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
= ap ⋅bp
⎛
sec
y = ∠ sec x = ∠ cos
k =1
p
log a N =
ctg
1.5
ap ⎛a⎞ ⎜ ⎟ = p b ⎝b⎠ = a
tg 0
cos
1 y = ∠ ctg x = ∠ tg x
k =1
n
q
q
+ an = ∑ ak
k
n
0 30 45 60 90
sen
y = ∠ tg x
∑ ⎣⎡ a + ( k − 1) d ⎦⎤ = 2 ⎣⎡ 2a + ( n − 1) d ⎦⎤
( a p ) = a pq
a
k =1
k =1
ap = a p−q aq
p/q
k
θ
n
n
n
a p ⋅ a q = a p+q
p
par
k =1
n
∑a
⎞ a n − k b k −1 ⎟ = a n − b n ∀ n ∈ ⎠ SUMAS Y PRODUCTOS k +1
n
= ∏ ak
k
impar
∑ c = nc
EXPONENTES
(a ⋅ b)
⎛
n
( a + b ) ⋅ ⎜ ∑ ( −1) a1 + a2 +
a ≥0y a =0 ⇔ a=0
k =1
⎛ n ⎞ k +1 ( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈ ⎝ k =1 ⎠ ⎝ k =1
a ≤ a y −a≤ a
ab = a b ó
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 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 =1 lim 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
Dx f ( x ) =
DERIVADAS f ( x + ∆x ) − f ( x ) df ∆y = 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
Jesús Rubí M. 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
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 1 d du ⋅ senh −1 u = dx 1 + u 2 dx -1 ±1 d du ⎪⎧+ si cosh u > 0 ⋅ , u >1 ⎨ cosh −1 u = -1 dx ⎪⎩− si cosh u < 0 u 2 − 1 dx 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
∫ { 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
a
b
b
b
a
a
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
n ≠ −1
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
u 1 ( 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 u 1 = − ∠ ctg a a du 1 u−a 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 du
∫
a2 − u2
2
= ∠ sen
= −∠ cos
2
2
du
∫
u 2 ± a2
(
u a
= ln u + u 2 ± a 2
udu =
du
a2 ± u2 du
(
∫ tg udu = tg u − u ∫ ctg udu = − ( ctg u + u ) 2
∫ u sen udu = sen u − u cos u ∫ u cos udu = cos u + u sen u INTEGRALES DE FUNCS TRIGO INV 2 ∫ ∠ 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
au
sen bu du =
2
+
f(
n)
( x0 )( x − x0 ) n!
f ( x ) = f ( 0) + f '( 0) x +
2
2
−1
= u∠ sec u − ∠ cosh u
∫ ∠ csc udu = u∠ csc u + ln ( u +
u 2 −1
) )
= u∠ csc u + ∠ cosh u INTEGRALES DE FUNCS HIP
2
2
f '' ( x0 )( x − x0 )
f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) + +
2
∫ 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
)
e au ( a sen bu − b cos bu ) a 2 + b2 e au ( a cos bu + b sen bu ) au ∫ e cos bu du = a 2 + b2 ALGUNAS SERIES
∫e
2
du ∫ u = ln u
)
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
∫u
u 1 − sen 2u 2 4 u 1 2 ∫ cos udu = 2 + 4 sen 2u
∫ sen
u a
+
+
f
( n)
( 0) x n!
2!
n
f '' ( 0 ) x 2!
: Taylor 2
n
: Maclaurin
x 2 x3 xn ex = 1 + x + + + + + n! 2! 3! x3 x 5 x 7 x 2 n −1 n −1 sen x = x − + − + + ( −1) 3! 5! 7! ( 2n − 1)! cos x = 1 −
x2 x4 x6 + − + 2! 4! 6!
+ ( −1)
n −1
x 2n−2
( 2n − 2 )!
n x 2 x3 x 4 n −1 x + − + + ( −1) n 2 3 4 2 n −1 x3 x5 x7 n −1 x ∠ tg x = x − + − + + ( −1) 3 5 7 2n − 1
ln (1 + x ) = x −
2
Formulario de Cálculo Diferencial e Integral ÁLGEBRA LINEAL Def. El determinante de una matriz a ⎤ ⎡a A = ⎢ 11 12 ⎥ ⎣ a21 a22 ⎦ está dado por
det A =
a11 a12 = a11a22 − a12 a21 . a21 a22
Def. El determinante de una matriz
⎡ a11 a12 ⎢ A = ⎢ a21 a22 ⎣⎢ a31 a32
a13 ⎤ ⎥ a23 ⎥ a33 ⎦⎥
está dado por
a11 a12 det A = a21 a22 a31
a32
a11 ⋅ a22 ⋅ a33 + a12 ⋅ a23 ⋅ a31 a13 a23 = + a13 ⋅ a21 ⋅ a32 − a11 ⋅ a23 ⋅ a32 . a33
−a12 ⋅ a21 ⋅ a33 − a13 ⋅ a22 ⋅ a31
Jesús Rubí M.
