Formulario de calculo integral y diferencial
3
Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. Fórmulas de Cálculo Diferencial e Integral VER.6.8 Jesús Rubí Miranda ([email protected] ) http://www.geocities.com/calculusjrm/ VALOR ABSOLUTO 1 1 1 1 si 0 si 0 y 0 y 0 0 ó ó n n k k k k n n k k k k a a a a a a a a a a a a a a ab ab a a a b a b a a = = = = ≥ ⎧ = ⎨ − < ⎩ =− ≤ − ≤ ≥ = ⇔ = = = + ≤ + ≤ ∏ ∏ ∑ ∑ EXPONENTES ( ) ( ) / p q p q p p q q q p pq p p p p p p q pq p a a a a a a a a ab a b a a b b a a + − ⋅ = = = ⋅ = ⋅ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ = LOGARITMOS 10 log log log log log log log log log log ln log log ln log log y log ln x a a a a a a a r a a b a b e N x a MN M N M M N N N r N N N N a a N N N N N = ⇒ = + = − = = = = = = ALGUNOS PRODUCTOS ad + ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 3 3 2 2 3 3 3 2 2 3 2 2 2 2 2 2 3 3 3 3 2 2 2 a c d ac a b a b a b a b a b a b a ab b a b a b a b a ab b x b x d x b dx bd ax b cx d acx ad bc x bd a b c d ac ad bc bd a b a ab ab b a b a ab ab b a b c a b c ab ac bc ⋅ + = + ⋅ − = − + ⋅ + = + = + + − ⋅ − = − = − + + ⋅ + = + + + + ⋅ + = + + + + ⋅ + = + + + + = + + + − = − + − + + = + + + + + 1 1 n n k k n n k a b a ab b a b a b a ab ab b a b a b a ab ab ab b a b a b a b a b n − − = − ⋅ + + = − − ⋅ + + + = − − ⋅ + + + + = − ⎛ ⎞ − ⋅ = − ∀ ∈ ⎜ ⎟ ⎝ ⎠ ∑ ` ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 5 4 3 2 2 3 4 5 6 6 a b a ab b a b a b a ab ab b a b a b a ab ab ab b a b a b a ab ab ab ab b a b + ⋅ − + = + + ⋅ − + − = − + ⋅ − + − + = + + ⋅ − + − + − = − ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 impar 1 par n k n k k n n k n k n k k n n k a b a b a b n a b a b a b n + − − = + − − = ⎛ ⎞ + ⋅ − = + ∀ ∈ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + ⋅ − = − ∀ ∈ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ` ` SUMAS Y PRODUCTOS n ( ) ( ) 1 2 1 1 1 1 1 1 1 1 0 n k k n k n n k k k k n n n k k k k k k k n k k n k a a a a c nc ca c a a b a b a a a a = = = = = = = − = + + + = = = + = + − = − ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ " ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 2 3 2 1 3 4 3 2 1 4 5 4 3 1 2 1 1 2 1 2 = 2 1 1 1 1 2 1 2 3 6 1 2 4 1 6 15 10 30 1 3 5 2 1 ! n k n n k k n k n k n k n k n k n a k d a n d n a l r a rl ar a r r k n n k n n n k n n n k n n n n n n n k n n k = − = = = = = = + − = + − ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ + − − = = − − = + = + + = + + = + + − + + + + − = = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ ∑ ∑ ∑ ∏ " ( ) ( ) 0 ! , !! n n n k k k k n n k k n x y x y k − = ≤ − ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ ∑ ( ) 1 2 1 2 1 2 1 2 ! ! ! ! k n n n n k k k n x x x x x x n n n + + + = ⋅ ∑ " " " CONSTANTES 9… 3.1415926535 2.71828182846 e π = = … TRIGONOMETRÍA 1 sen csc sen 1 cos sec cos sen 1 tg ctg cos tg CO HIP CA HIP CO CA θ θ θ θ θ θ θ θ θ θ θ = = = = = = = radianes=180 π D CA CO HIP θ θ sin cos tg ctg sec csc 0 D 0 1 0 ∞ 1 ∞ 30 D 12 3 2 1 3 3 2 3 2 45 D 1 2 1 2 1 1 2 2 60 D 3 2 12 3 1 3 2 2 3 90 D 1 0 ∞ 0 ∞ 1 [ ] [ ] sin , 2 2 cos 0, tg , 2 2 1 ctg tg 0, 1 sec cos 0, 1 csc sen , 2 2 y x y y x y y x y y x y x y x y x y x y x π π π π π π π π π ⎡ ⎤ =∠ ∈− ⎢ ⎥ ⎣ ⎦ =∠ ∈ =∠ ∈ − =∠ =∠ ∈ =∠ =∠ ∈ ⎡ ⎤ =∠ =∠ ∈− ⎢ ⎥ ⎣ ⎦ Gráfica 1. Las funciones trigonométricas: sin x , cos x , tg x : -8 -6 -4 -2 0 2 4 6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 sen x cos x tg x Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x : -8 -6 -4 -2 0 2 4 6 8 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 csc x sec x ctg x Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctg x : -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 arc sen x arc cos x arc tg x Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : -5 0 -2 -1 0 1 2 3 4 arc ctg x arc sec x arc csc x IDENTIDADES TRIGONOMÉTRICAS 2 2 2 2 2 2 sin cos 1 1 ctg csc tg 1 sec θ θ θ θ θ θ + = + = + = ( ) ( ) ( ) sin sin cos cos tg tg θ θ θ θ θ θ − =− − = − =− ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin 2 sin cos 2 cos tg 2 tg sin sin cos cos tg tg sin 1 sin cos 1 cos tg tg n n n n n θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ + = + = + = + =− + =− + = + =− + =− + = ( ) ( ) ( ) ( ) ( ) sin 0 cos 1 tg 0 2 1 sin 1 2 2 1 cos 0 2 2 1 tg 2 n n n n n n n n π π π π π π = =− = + ⎛ ⎞ =− ⎜ ⎟ ⎝ ⎠ + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ + ⎛ ⎞ =∞ ⎜ ⎟ ⎝ ⎠ sin cos 2 cos sin 2 π θ θ π θ θ ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 sin sin cos cos sin cos cos cos sin sin tg tg tg 1 tg tg sin 2 2sin cos cos 2 cos sin 2 tg tg 2 1 tg 1 sin 1 cos 2 2 1 cos 1 cos 2 2 1 cos 2 tg 1 cos 2 α β α β α β α β α β α β α β α β α β θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ± = ± ± = ± ± = = = − = − = − = + − = + ∓ ∓ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 sin sin 2sin cos 2 2 1 1 sin sin 2sin cos 2 2 1 1 cos cos 2 cos cos 2 2 1 1 cos cos 2sin sin 2 2 α β α β α β α β α β α β α β α β α β α β α β α β + = + ⋅ − − = − ⋅ + + = + ⋅ − − =− + ⋅ − ( ) sin tg tg cos cos α β α β α β ± ± = ⋅ ( ) ( ) ( ) ( ) ( ) ( ) 1 sin cos sin sin 2 1 sin sin cos cos 2 1 cos cos cos cos 2 α β α β α β α β α β α β α β α β α β ⋅ = − + + ⎡ ⎤ ⎣ ⎦ ⋅ = − − + ⎡ ⎤ ⎣ ⎦ ⋅ = − + + ⎡ ⎤ ⎣ ⎦ tg tg tg tg ctg ctg α β α β α β + ⋅ = + FUNCIONES HIPERBÓLICAS sinh 2 cosh 2 sinh tgh cosh 1 ctgh tgh 1 2 sech cosh 1 2 csch sinh x x x x x x x x x x x x x e e x x e e x x e e e e x x e e x x e e x x e e − − − − − − − = + = − = = + + = = − = = + = = − x x e e − − x x [ {} ] {} {} sinh : cosh : 1, tgh : 1,1 ctgh : 0 , 1 1, sech : 0,1 csch : 0 0 → → ∞ →− − → −∞ − ∪ ∞ → − → − \ \ \ \ \ \ \ \ Gráfica 5. Las funciones hiperbólicas sinh x , cosh x , tgh x : -5 0 -4 -3 -2 -1 0 1 2 3 4 5 senh x cosh x tgh x FUNCIONES HIPERBÓLICAS INV ( ) ( ) 1 2 1 2 1 1 2 1 2 1 sinh ln 1, cosh ln 1, 1 1 1 tgh ln , 1 2 1 1 1 ctgh ln , 1 2 1 1 1 sech ln , 0 1 1 1 csch ln , 0 x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − − = + + ∀∈ = ± − ≥ + ⎛ ⎞ = < ⎜ ⎟ − ⎝ ⎠ + ⎛ ⎞ = > ⎜ ⎟ − ⎝ ⎠ ⎛ ⎞ ± − ⎜ ⎟ = < ≤ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + ⎜ ⎟ = + ≠ ⎜ ⎟ ⎝ ⎠ \
-
Upload
angel-gomez -
Category
Education
-
view
531 -
download
6
Transcript of Formulario de calculo integral y diferencial
Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
Fórmulas de Cálculo Diferencial e Integral VER.6.8 Jesús Rubí Miranda ([email protected]) http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO
1 1
1 1
si 0 si 0
y
0 y 0 0
ó
ó
n n
k kk k
n n
k kk k
a aa
a a
a a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥⎧= ⎨− <⎩= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
EXPONENTES
( )( )
/
p q p q
pp q
q
qp pq
p p p
p p
p
qp q p
a a aa aa
a a
a b a b
a ab b
a a
+
−
⋅ =
=
=
⋅ = ⋅
⎛ ⎞ =⎜ ⎟⎝ ⎠
=
LOGARITMOS
10
loglog log log
log log log
log loglog lnloglog ln
log log y log ln
xa
a a a
a a a
ra a
ba
b
e
N x aMN M NM M NNN r N
N NNa a
N
N N N N
= ⇒
= +
= −
=
= =
= =
=
ALGUNOS PRODUCTOS ad+( )
( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( )( )
2 2
2 2 2
2 2 2
2
2
3 3 2 2 3
3 3 2 2 3
2 2 2 2
2
2
3 3
3 3
2 2 2
a c d ac
a b a b a b
a b a b a b a ab b
a b a b a b a ab b
x b x d x b d x bd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
a b a a b ab b
a b a a b ab b
a b c a b c ab ac bc
⋅ + =
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + + + +
1
1
n
n k k n n
k
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a b a b n− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
⎛ ⎞− ⋅ = − ∀ ∈⎜ ⎟⎝ ⎠∑
( ) ( )( ) ( )( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
( ) ( )( ) ( )( ) ( )( ) ( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
5 4 3 2 2 3 4 5 6 6
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a a b a b a b ab b a b
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
( ) ( )
( ) ( )
1 1
1
1 1
1
1 impar
1 par
nk n k k n n
k
nk n k k n n
k
a b a b a b n
a b a b a b n
+ − −
=
+ − −
=
⎛ ⎞+ ⋅ − = + ∀ ∈⎜ ⎟⎝ ⎠⎛ ⎞+ ⋅ − = − ∀ ∈⎜ ⎟⎝ ⎠
∑
∑ SUMAS Y PRODUCTOS
n
( )
( )
1 21
1
1 1
1 1 1
1 0
n kk
n
k
n n
k kk kn n n
k k k kk k kn
k k nk
a a a a
c nc
ca c a
a b a b
a a a a
=
=
= =
= = =
−=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
( )
1
( )
( )
( )
( )
( )
( )( )
1
1
1
2
1
2 3 2
1
3 4 3 2
1
4 5 4 3
12
1
1 2 12
=2
11 1
121 2 361 241 6 15 10
301 3 5 2 1
!
n
k
nnk
k
n
kn
kn
kn
k
n
k
na k d a n d
n a l
r a rlar ar r
k n n
k n n n
k n n n
k n n n n
n n
n k
n nk
=
−
=
=
=
=
=
=
+ − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
+
− −= =
− −
= +
= + +
= + +
= + + −
+ + + + − =
=
⎛ ⎞=⎜ ⎟
⎝ ⎠
∑
∑
∑
∑
∑
∑
∏
( )
( )0
! , ! !
nn n k k
k
k nn k k
nx y x y
k−
=
≤−
⎛ ⎞+ = ⎜ ⎟
⎝ ⎠∑
( ) 1 21 2 1 2
1 2
!! ! !
kn nn n
k kk
nx x x x x x n n n
+ + + = ⋅∑CONSTANTES 9…3.1415926535
2.71828182846eπ == …
TRIGONOMETRÍA
1sen cscsen
1cos seccos
sen 1tg ctgcos tg
COHIPCAHIP
COCA
θ θθ
θ θθ
θθ θθ θ
= =
= =
= = =
radianes=180π
CA
COHIP
θ
θ sin cos tg ctg sec csc0 0 1 0 ∞ 1 ∞30 1 2 3 2 1 3 3 2 3 245 1 2 1 2 1 1 2 260 3 2 1 2 3 1 3 2 2 390 1 0 ∞ 0 ∞ 1
[ ]
[ ]
sin ,2 2
cos 0,
tg ,2 2
1ctg tg 0,
1sec cos 0,
1csc sen ,2 2
y x y
y x y
y x y
y x yx
y x yx
y x yx
π π
π
π π
π
π
π π
⎡ ⎤= ∠ ∈ −⎢ ⎥⎣ ⎦= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
⎡ ⎤= ∠ = ∠ ∈ −⎢ ⎥⎣ ⎦
Gráfica 1. Las funciones trigonométricas: sin x , cos x , tg x :
-8 -6 -4 -2 0 2 4 6 8-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sen xcos xtg x
Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x :
-8 -6 -4 -2 0 2 4 6 8-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
csc xsec xctg x
Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctg x :
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
arc sen xarc cos xarc tg x
Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x :
-5 0 5-2
-1
0
1
2
3
4
arc ctg xarc sec xarc csc x
IDENTIDADES TRIGONOMÉTRICAS
2 2
2 2
2 2
sin cos 11 ctg csctg 1 sec
θ θθ θ
θ θ
+ =
+ =
+ =
( )( )( )
sin sin
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
( )( )( )( )( )( )( ) ( )( ) ( )( )
sin 2 sin
cos 2 cos
tg 2 tg
sin sin
cos cos
tg tg
sin 1 sin
cos 1 cos
tg tg
n
n
n
n
n
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( )( ) ( )( )
( )
sin 0
cos 1
tg 0
2 1sin 12
2 1cos 02
2 1tg2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+⎛ ⎞ = −⎜ ⎟⎝ ⎠
+⎛ ⎞ =⎜ ⎟⎝ ⎠
+⎛ ⎞ = ∞⎜ ⎟⎝ ⎠
sin cos2
cos sin2
πθ θ
πθ θ
⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠
( )( )
( )
( )
( )
2 2
2
2
2
2
sin sin cos cos sin
cos cos cos sin sintg tgtg
1 tg tgsin 2 2sin coscos 2 cos sin
2 tgtg 21 tg1sin 1 cos 221cos 1 cos 22
1 cos 2tg1 cos 2
α β α β α β
α β α β α βα βα βα β
θ θ θθ θ θ
θθθ
θ θ
θ θ
θθθ
± = ±
± =
±± =
=
= −
=−
= −
= +
−=
+
∓
∓
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1sin sin 2sin cos2 21 1sin sin 2sin cos2 21 1cos cos 2cos cos2 21 1cos cos 2sin sin2 2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
( )sintg tg
cos cosα β
α βα β
±± =
⋅
( ) ( )
( ) ( )
( ) ( )
1sin cos sin sin21sin sin cos cos21cos cos cos cos2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
tg tgtg tgctg ctg
α βα βα β+
⋅ =+
FUNCIONES HIPERBÓLICAS
sinh2
cosh2
sinhtghcosh
1ctghtgh
1 2sechcosh
1 2cschsinh
x x
x x
x
x
x x
x x
x x
x
e ex
x e exx e e
e exx e e
xx e e
xx e e
−
−
−
−
−
−
−
=
+=
−= =
++
= =−
= =+
= =−
x xe e−−
x
x
[
]
sinh :cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
Gráfica 5. Las funciones hiperbólicas sinh x ,
cosh x , tgh x :
-5 0 5-4
-3
-2
-1
0
1
2
3
4
5
senh xcosh xtgh x
FUNCIONES HIPERBÓLICAS INV
( )( )
1 2
1 2
1
1
21
21
sinh ln 1 ,
cosh ln 1 , 1
1 1tgh ln , 12 11 1ctgh ln , 12 1
1 1sech ln , 0 1
1 1csch ln , 0
x x x x
x x x x
xx xx
xx xx
xx xx
xx xx x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞= <⎜ ⎟−⎝ ⎠+⎛ ⎞= >⎜ ⎟−⎝ ⎠
⎛ ⎞± −⎜ ⎟= < ≤⎜ ⎟⎝ ⎠⎛ ⎞+⎜ ⎟= + ≠⎜ ⎟⎝ ⎠
Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
IDENTIDADES DE FUNCS HIP 2 2sinh 1x x
( )( )( )
2 2
2 2
cosh1 tgh sechctgh 1 cschsinh sinh
cosh cosh
tgh tgh
x xx xx x
x x
x x
− =
− =
− = −
− =
− = −
− =
( )( )
( )
2 2
2
sinh sinh cosh cosh sinh
cosh cosh cosh sinh sinhtgh tghtgh
1 tgh tghsinh 2 2sinh coshcosh 2 cosh sinh
2 tghtgh 21 tgh
x y x y x y
x y x y x yx yx y
x yx x xx x x
xxx
± = ±
± = ±
±± =
±=
= +
=+
( )
( )
2
2
2
1sinh cosh 2 121cosh cosh 2 12
cosh 2 1tghcosh 2 1
x x
x x
xxx
= −
= +
−=
+
sinh 2tghcosh 2 1
xxx
=+
cosh sinhcosh sinh
x
x
e x xe x x−
= +
= − OTRAS
( ) ( )
2
2
2
0
4 2
4 discriminanteexp cos sin si ,
ax bx c
b b acxa
b aci e iαα β β β α β
+ + =
− ± −⇒ =
− =
± = ± ∈
LÍMITES
( )1
0
0
0
0
1
lim 1 2.71828...
1lim 1
senlim 1
1 coslim 0
1lim 1
1lim 1ln
xx
x
x
x
x
x
x
x
x e
ex
xx
xx
ex
xx
→
→∞
→
→
→
→
+ = =
⎛ ⎞+ =⎜ ⎟⎝ ⎠
=
−=
−=
−=
DERIVADAS
( ) ( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x x x
n n
f x x f xdf yD f xdx x x
d cdxd cx cdxd cx ncxdxd du dv dwu v wdx dx dx dxd ducu cdx dx
∆ → ∆ →
−
+ ∆ − ∆= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
( )
( )
( ) ( )
( )
2
1n n
d dv duuv u vdx dx dxd dw dv duuvw uv uw vwdx dx dx dx
v du dx u dv dxd udx v vd duu nudx dx
−
= +
= + +
−⎛ ⎞ =⎜ ⎟⎝ ⎠
=
( )( )
( )( )
12
1 2
(Regla de la Cadena)
1
donde
dF dF dudx du dxdudx dx du
dF dudFdx dx du
x f tf tdy dtdydx dx dt f t y f t
= ⋅
=
=
=⎧′ ⎪= = ⎨′ =⎪⎩
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( ) 1
ln
loglog
loglog 0, 1
ln
ln
aa
u u
u u
v v v
udx u u dxd e duudx u dx
ed duu adx u dxd due edx dxd dua a adx dxd du dvu vu u udx dx dx
−
= = ⋅
= ⋅
= ⋅ >
= ⋅
= ⋅
= + ⋅ ⋅
1d du dx du
a ≠
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
sin cos
cos sin
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
u udx dxd duu udx dxd duu udx dxd duu udx dxd duu u udx dxd duu u udx dxd duu udx dx
=
= −
=
= −
=
= −
=
d du
DERIV DE FUNCS TRIGO INVER
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
sin1
1cos1
1tg1
1ctg1
si 11secsi 11si 11cscsi 11
1vers2
udx dxud duudx dxud duudx dxud duudx dxu
ud duuudx dxu uud duuudx dxu u
d duudx dxu u
∠ = ⋅−
∠ = − ⋅−
∠ = ⋅+
∠ = − ⋅+
+ >⎧∠ = ± ⋅ ⎨
1d du
− < −⎩−− >⎧
∠ = ⋅ ⎨+ < −⎩−
∠ = ⋅−
∓
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
sinh cosh
cosh sinh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
u udx dxd duu udx dxd duu udx dxd duu udx dxd duu u udx dxd duu u udx dx
=
=
=
= −
= −
= −
d du
DERIVADA DE FUNCS HIP INV 1
2
-11
-12
12
12
11
12
senh1
si cosh 01cosh , 1 si cosh 01
1tgh , 11
1ctgh , 11
si sech 0, 0,11sechsi sech 0, 0,11
udx dxu
ud duu udx dx uud duu udx u dxd duu udx u dx
u ud duudx dx u uu u
−
−
−
−
−−
−
= ⋅+
⎧+ >± ⎪= ⋅ > ⎨− <− ⎪⎩
= ⋅ <−
= ⋅ >−
⎧− > ∈⎪= ⋅ ⎨+ < ∈− ⎩
∓
1d du
1
2
1csch , 01
d duu udx dxu u
−
⎪
= − ⋅ ≠+
INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de integración).
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )( ) [ ]
( ) ( )( ) ( ) [ ]
( ) ( )
0
, , ,
,
si
b b b
a a ab b
a ab c b
a a cb a
a ba
ab
a
b b
a a
b b
a a
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
m b a f x dx M b a
m f x M x a b m M
f x dx g x dx
f x g x x a b
f x dx f x dx a b
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
≤ <
∫ ∫ ∫∫ ∫∫ ∫ ∫∫ ∫∫
∫
∫ ∫
∫ ∫
INTEGRALES
( ) ( )
( )
( )1
Integración por partes
11
ln
nn
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
uu du nn
du uu
+
=
=
± ± ± = ± ± ±
= −
= ≠ −+
=
∫∫ ∫∫ ∫ ∫ ∫∫ ∫
∫
∫
INTEGRALES DE FUNCS LOG & EXP
( )
( )
( ) ( )
( )
( )
2
2
01ln
1ln ln
1
ln ln ln 1
1log ln ln 1ln ln
log 2log 14
ln 2ln 14
u u
uu
uu
u u
a
a a
e du e
aaa duaa
aua du ua a
ue du e u
udu u u u u u
uudu u u u ua auu udu u
uu udu u
=
>⎧= ⎨ ≠⎩
⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠
= −
= − = −
= − = −
= ⋅ −
= −
∫
∫
∫
∫∫
∫
∫
∫
INTEGRALES DE FUNCS TRIGO
2
2
sin cos
cos sin
sec tg
csc ctg
sec tg sec
csc ctg csc
udu u
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫∫∫∫∫∫
tg ln cos ln sec
ctg ln sin
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫∫∫∫
( )
2
2
2
2
1sin sin 22 4
1cos sin 22 4
tg tg
ctg ctg
uudu u
uudu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫∫∫
sin sin cos
cos cos sin
u udu u u u
u udu u u u
= −
= +
∫∫
INTEGRALES DE FUNCS TRIGO INV
( )
( )
2
2
2
2
2
2
sin sin 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
csc cosh
udu u u u
udu u u u
udu u u u
udu u u u
udu u u u u
u u u
udu u u u u
u u u
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − +
= ∠ −∠
∠ = ∠ + + −
= ∠ +∠
∫∫∫∫∫
∫
−
INTEGRALES DE FUNCS HIP
2
2
sinh cosh
cosh sinh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu u
udu u
udu u
udu u
u udu u
u udu u
=
=
=
= −
= −
= −
∫∫∫∫∫∫
( )( )1
tgh ln cosh
ctgh ln sinh
sech tg sinh
csch ctgh cosh
1 ln tgh2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫∫∫∫
INTEGRALES DE FRAC
( )
( )
2 2
2 22 2
2 22 2
tg
1 ctg
1 ln 21 ln
2
duu a a a
ua a
du u a u au a a u a
du a u u aa u a a u
= ∠+
= − ∠
−= >
− ++
= <− −
∫
∫
∫
1 u
INTEGRALES CON
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
22 2 2 2
22 2 2 2 2 2
sin
cos
ln
1 ln
1 cos
1 sec
sen2 2
ln2 2
du uaa u
ua
du u u au a
du uau a u a a u
du aa uu u a
ua au a ua u du a u
au au a du u a u u a
= ∠−
= −∠
= + ±±
=± + ±
= ∠−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
MÁS INTEGRALES ( )
( )2 2
2 2
3
sin cossin
cos sincos
1 1sec sec tg ln sec tg2 2
au
auau
e a bu b bue bu du
a be a bu b bu
e bu dua b
u du u u u u
−=
++
=+
= + +
∫
∫
∫
au
ALGUNAS SERIES
( ) ( ) ( ) ( ) ( )( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
20 0
0 0 0
0 0
2
2 3
3 5 7 2 11
2 4 6
'''
2!
: Taylor!
'' 00 ' 0
2!0
: Maclaurin!
12! 3! !
sin 13! 5! 7! 2 1 !
cos 12! 4!
nn
n n
nx
nn
f x x xf x f x f x x x
f x x xn
f xf x f f x
f xn
x x xe xn
x x x xx xn
x x xx
−−
−= + − +
−+ +
= + +
+ +
= + + + + + +
= − + − + + −−
= − + − ( ) ( )
( ) ( )
( )
2 21
2 3 41
3 5 7 2 11
16! 2 2 !
ln 1 12 3 4
tg 13 5 7 2 1
nn
nn
nn
xn
x x x xx xn
x x x xx xn
−−
−
−−
+ + −−
+ = − + − + + −
∠ = − + − + + −−
Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. ALFABETO GRIEGO
ayúscula Minúscula Nombre M Equivalente Romano
1 Α α Alfa A 2 Β β Beta B 3 Γ γ Gamma G 4 ∆ δ Delta D 5 Ε ε Epsilon E 6 Ζ ζ Zeta Z 7 Η η Eta H 8 Θ θ ϑ Teta Q 9 Ι ι Iota I 10 Κ κ Kappa K 11 Λ λ Lambda L 12 Μ µ Mu M 13 Ν ν Nu N 14 Ξ ξ Xi X 15 Ο ο Omicron O 16 Π π ϖ Pi P 17 Ρ ρ Rho R 18 Σ σ ς Sigma S 19 Τ τ Tau T 20 Υ υ Ipsilon U 21 Φ φ ϕ Phi F 22 Χ χ Ji C 23 Ψ ψ Psi Y 24 Ω ω Omega W
NOTACIÓN
Seno. sincos Coseno. tg Tangente. sec Secante. csc Cosecante. ctg Cotangente. vers Verso seno.
arcsin sinθ 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.
Conjunto de enteros. , 2, 1,0,1,2,= − −… …
Conjunto de números racionales. c Conjunto de números irracionales.
1,2,3,= … Conjunto de números naturales.
Conjunto de números complejos.
