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
π radianes=180
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
IDENTIDADES TRIGONOMÉTRICAS
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
=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
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
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
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
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
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 =
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
2
du
a2 ± u2
du
+
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
)
+
f(
n)
( x0 )( x − x0 )
n!
2!
n
: Taylor
f '' ( 0 ) x 2
f ( x ) = f ( 0) + f '( 0) x +
2!
( 0 ) x n : Maclaurin
n!
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 −
+
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)
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 −
)
f '' ( x0 )( x − x0 )
f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) +
2
2
(
= ln u + u 2 ± a 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
du
u 2 ± a2
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