formulario de derivadas para calculo integral
TRANSCRIPT
FORMULARIO DE CÁLCULO DIFERENCIAL E INTEGRAL VER.3.6 Jesús Rubí Miranda ([email protected]) http://mx.geocities.com/estadisticapapers/ http://mx.geocities.com/dicalculus/ 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 aa a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥= − <= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
( )( )
/
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
+
−
⋅ =
=
=
⋅ = ⋅
=
=
EXPONENTES
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 a NMN M NM M NNN r N
N NNa a
N N N N
= ⇒ == +
= −
=
= =
= =
LOGARITMOS
ALGUNOS PRODUCTOS
( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( )( )
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
3 3
3 3
a c d ac ad
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
⋅ + = +
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + 2 2 2 2ab ac bc+ + +
( ) ( )( ) ( )( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5
1
1
nn 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− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
− ⋅ = − ∀ ∈ ∑
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
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
CA
CO HIP
θ
Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : ( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1sen sen 2sen cos2 21 1sen sen 2sen cos2 21 1cos cos 2cos cos2 2
1 1cos cos 2sen sen2 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
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
-5 0 5-2
-1
0
1
2
3
4
arc ctg xarc sec xarc csc x
( ) ( )
( ) ( )
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
+ − −
=
+ − −
=
+ ⋅ − = + ∀ ∈ + ⋅ − = − ∀ ∈
∑
∑
( )
( )
1 21
1
1 1
1 1 1
1 01
n
n kk
n
kn 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
=
=
= =
= = =
−=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
SUMAS Y PRODUCTOS
n
θ sen cos tg ctg sec csc 0 1 0 ∞ 1 ∞ 1 2 3 2 1 3 3 2 3 2 1 2 1 2 1 1 2 2
60 3 2 1 2 3 1 3 2 2 3 90 0 1 ∞ 0 ∞
0 ( )sentg tg
cos cosα β
α βα β
±± =
⋅
3045 ( ) ( )
( ) ( )
( ) ( )
1sen cos sen sen21sen sen cos cos21cos cos cos cos2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +
⋅ = − − +
⋅ = − + +
2 2
2 2
2 2
sen cos 11 ctg csctg 1 sec
θ θ
θ θ
θ θ
+ =
+ =
+ =
IDENTIDADES TRIGONOMÉTRICAS
1
[ ]
[ ]
sen ,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
= ∠ ∈ − = ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
= ∠ = ∠ ∈ −
tg tgtg tgctg ctg
α βα β
α β+
⋅ =+
( )( )( )
sen sen
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
senh
2
cosh2
senhtghcosh
1ctghtgh
1 2sechcosh
1 2cschsenh
x x
x x
x x
x x
x x
x x
x x
x x
e ex
e ex
x e exx e e
e exx e e
xx e e
xx e e
−
−
−
−
−
−
−
−
−=
+=
−= =
++
= =−
= =+
= =−
FUNCIONES HIPERBÓLICAS
( )( )( )( )( )( )( ) ( )( ) ( )( )
sen 2 sen
cos 2 cos
tg 2 tg
sen sen
cos cos
tg tg
sen 1 sen
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
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( ) ( )
( )
( )
( )
( )
( )( )
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 1030
1 3 5 2 1
!
k
nnk
kn
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−
=
≤−
+ =
∑
Gráfica 1. Las funciones trigonométricas: , cos x , tg x :
sen 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
[
{ }]
{ } { }
senh :cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
( )( ) ( )( )
( )
sen 0
cos 1
tg 0
2 1sen 12
2 1cos 02
2 1tg2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+ = −
+ =
+ = ∞
Gráfica 2. Las funciones trigonométricas , sec x , ctg x :
csc x
Gráfica 5. Las funciones hiperbólicas ,
, : senh x
cosh x tgh x
( ) 1 21 2 1 2
1 2
!! ! !
kn nn n
k kk
nx x x x x xn n n
+ + + = ⋅∑
-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
-5 0 5
-4
-3
-2
-1
0
1
2
3
4
5
senh xcosh xtgh x
sen cos2
cos sen2
πθ θ
πθ θ
= − = +
( )
3.141592653592.71828182846e
π ==
CONSTANTES……
( )
( )
( )
( )
2 2
2
2
2
2
sen sen cos cos sen
cos cos cos sen sentg tgtg
1 tg tgsen 2 2sen coscos 2 cos sen
2 tgtg 21 tg
1sen 1 cos 221cos 1 cos 22
1 cos 2tg1 cos 2
α β α β α β
α β α β α β
α βα β
α βθ θ θ
θ θ θθ
θθ
θ θ
θ θ
θθ
θ
± = ±
± =
±± =
=
= −
=−
= −
= +
−=
+
∓
∓
1sen csc
sen1cos sec
cossen 1tg ctgcos tg
COHIPCAHIP
COCA
θ θθ
θ θθ
θθ θ
θ θ
= =
= =
= = =
TRIGONOMETRÍA
Gráfica 3. Las funciones trigonométricas inversas arcsen 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
radianes=180π
( )( )
1 2
1 2
1
1
21
21
senh 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
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+ = < − + = > −
± − = < ≤ + = + ≠
FUNCS HIPERBÓLICAS INVERSAS
( ) ( ) ( ) ( ) ( )( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
20 0
0 0 0
0 0
2
2 3
3 5 7 2 11
'''
2!
: Taylor!
'' 00 ' 0
2!0
: Maclaurin!
12! 3! !
sen 13! 5! 7! 2 1 !
cos
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
−−
−= + − +
−+ +
= + +
+ +
= + + + + + +
= − + − + + −−
=
ALGUNAS SERIES
( ) ( )
( ) ( )
( )
2 4 6 2 21
2 3 41
3 5 7 2 11
1 12! 4! 6! 2 2 !
ln 1 12 3 4
tg 13 5 7 2 1
nn
nn
nn
x x x xn
x x x xx xn
x x x xx xn
−−
−
−−
− + − + + −−
+ = − + − + + −
∠ = − + − + + −−
( )
( )2 2
2 2
sen cossen
cos sencos
auau
auau
e a bu b bue bu du
a be a bu b bu
e bu dua b
−=
++
=+
∫
∫
MAS INTEGRALES
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
22 2 2 2
22 2 2 2 2 2
sen
cos
ln
1 ln
1 cos
1 sec
sen2 2
ln2 2
du uaa uua
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
= ∠−
= −∠
= + ±±
=± + ±
= ∠−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
INTEGRALES CON
( )
( )
2 2
2 22 2
2 22 2
1 tg
1 ctg
1 ln 21 ln
2
du ua au a
ua a
du u a u aa u au a
du a u u aa a ua u
= ∠+
= − ∠
−= >
+−+
= <−−
∫
∫
∫
INTREGRALES DE FRAC
( )( )1
tgh ln cosh
ctgh ln senh
sech tg senh
csch ctgh cosh
1 ln tgh2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫∫∫∫
( )( )( )
2 2
2 2
2
cosh senh 11 tgh sechctgh 1 cschsenh senh
cosh cosh
tgh tgh
x xx x
x xx x
x x
x x
− =
− =
− =
− = −
− =
− = −
IDENTIDADES DE FUNCS HIP
( )( )
( )
2 2
2
senh senh cosh cosh senh
cosh cosh cosh senh senhtgh tghtgh
1 tgh tghsenh 2 2senh coshcosh 2 cosh senh
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
1senh cosh 2 121cosh cosh 2 12
cosh 2 1tghcosh 2 1
x x
x x
xxx
= −
= +
−=
+
senh 2tghcosh 2 1
xxx
=+
2
2
2
0
4 2
4 discriminante
ax bx c
b b acxa
b ac
+ + =
− ± −⇒ =
− =
OTRAS
( )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
→
→∞
→
→
→
→
+ = =
+ =
=
−=
−=
−=
LÍMITES
( ) ( ) ( )
( )
( )
( )
( )
( )
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
∆ → ∆ →
−
+ ∆ − ∆= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
DERIVADAS
( )
( )
( ) ( )
( )
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
= ⋅
=
=
=′ = = ′ =
( )
( )
( )
( )
( )
( ) 1
1ln
loglog
loglog 0, 1
ln
ln
aa
u u
u u
v v v
du dxd duudx u u dxd e duudx u dx
ed duu a adx u dxd due edx dxd dua a adx dxd du dvu vu u udx dx dx
−
= = ⋅
= ⋅
= ⋅ > ≠
= ⋅
= ⋅
= + ⋅ ⋅
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( )
( )
2
2
sen cos
cos sen
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
d duu udx dxd duu udx dxd duu udx dxd duu udx dxd duu u udx dxd duu u udx dxd duu udx dx
=
= −
=
= −
=
= −
=
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
1sen1
1cos1
1tg1
1ctg1
si 11sec si 11 si 11csc si 11
1vers2
d duudx dxud duudx dxud duudx dxud duudx dxu
ud duuudx dxu uud duuudx dxu u
d udx u u
∠ = ⋅−
∠ = − ⋅−
∠ = ⋅+
∠ = − ⋅+
+ >∠ = ± ⋅ − < −−
− >∠ = ⋅ + < −−
∠ =−
DERIV DE FUNCS TRIGO INVER
∓
2
dudx⋅
2
2
senh cosh
cosh senh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
d duu udx dxd duu udx dxd duu udx dxd duu udx dxd duu u udx dxd duu u udx dx
=
=
=
= −
= −
= −
DERIVADA DE FUNCS HIPERBÓLICAS
1
2
-11
-12
12
12
11
2
1senh1
si cosh 01cosh , 1 si cosh 01
1tgh , 11
1ctgh , 11
si sech 0, 0,1sech1
d duudx dxu
ud duu udx dx uud duu udx dxud duu udx dxu
u ud duudx dxu u
−
−
−
−
−−
= ⋅+
+ >± = ⋅ > − <−
= ⋅ <−
= ⋅ >−
− > ∈= ⋅
−
DERIVADA DE FUNCS HIP INV
∓1
1
2
1
si sech 0, 0,1
1csch , 01
u u
d duu udx dxu u
−
−
+ < ∈
= − ⋅ ≠+
( )( )
( ) ( )
( )
( )
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 LOG & EXP
( ) ( ){ } ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )( ) [ ]
( ) ( )( ) ( )
0
, , ,
,
b b b
a a ab b
a ab c b
a a cb a
a ba
ab
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
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
∫ ∫ ∫∫ ∫∫ ∫ ∫∫ ∫∫
∫
∫ ∫
INTEGRALES DEFINIDAS,PROPIEDADES
[ ]
( ) ( ) si b b
a a
b
f x dx f x dx a b≤ <∫ ∫
( ) ( )( )
( )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
∫
2
2
sen cos
cos sen
sec tg
csc ctg
sec tg sec
csc ctg csc
udu u
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫∫∫∫∫∫
INTEGRALES DE FUNCS TRIGO
tg ln cos ln sec
ctg ln sen
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫∫∫∫
( )
2
2
2
2
1sen sen 22 4
1cos sen 22 4
tg tg
ctg ctg
uudu u
uudu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫∫∫
sen sen cos
cos cos sen
u udu u u u
u udu u u u
= −
= +
∫∫
INTEGRALES DE FUNCS
( )
( )
2
2
2
2
2
2
sen sen 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
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
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − + −
= ∠ −∠
∠ = ∠ + + −
∫∫∫∫∫
∫
TRIGO INV
csc coshu u u= ∠ +∠INTEGRALES DE FUNCS HIP
2
2
senh cosh
cosh senh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu u
udu u
udu u
udu u
u udu u
u udu u
=
=
=
= −
= −
= −
∫∫∫∫∫∫