Embed Size (px)
Citation preview
Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
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
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 ac 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
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
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− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
⎛ ⎞− ⋅ = − ∀ ∈⎜ ⎟⎝ ⎠∑
( ) ( )( ) ( )( ) ( )
( )
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
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
=
=
= =
= = =
−=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
( )
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 sen 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
[ ]
[ ]
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
⎡ ⎤= ∠ ∈ −⎢ ⎥⎣ ⎦= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
⎡ ⎤= ∠ = ∠ ∈ −⎢ ⎥⎣ ⎦
Gráfica 1. Las funciones trigonométricas: sen 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 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
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
2cos 1θ θ2
2 2
2 2
sen1 ctg csctg 1 sec
θ θθ θ
+ =
+ =
+ =
( )( )( )
sen sen
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
( )( )( )( )( )( )( ) ( )( ) ( )( )
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
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( )( ) ( )( )
( )
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
=
= −
=
+⎛ ⎞ = −⎜ ⎟⎝ ⎠
+⎛ ⎞ =⎜ ⎟⎝ ⎠
+⎛ ⎞ = ∞⎜ ⎟⎝ ⎠
sen cos2
cos sen2
πθ θ
πθ θ
⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠
( )( )
( )
( )
( )
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 βα βα βα β
θ θ θθ θ θ
θθθ
θ θ
θ θ
θθθ
± = ±
± =
±± =
=
= −
=−
= −
= +
−=
+
∓
∓
( ) ( )
( ) ( )
( ) ( )
( ) ( )
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
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
( )sentg tg
cos cosα β
α βα β
±± =
⋅
( ) ( )
( ) ( )
( ) ( )
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
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
tg tgtg tgctg ctg
α βα βα β+
⋅ =+
FUNCIONES HIPERBÓLICAS
senh2
cosh2
senhtghcosh
1ctghtgh
1 2sechcosh
1 2cschsenh
x x
x x
x
x
x x
x x
x x
x
e ex
x e exx e e
e exx e e
x
x xe e
x
x
x e e
xx e e
−
−
−
−
−
−
−
=
+=
−= =
++
= =−
= =+
= =−
−−
[
{ }]
{ } { }
senh :cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
Gráfica 5. Las funciones hiperbólicas senh 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
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
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞= <⎜ ⎟−⎝ ⎠+⎛ ⎞= >⎜ ⎟−⎝ ⎠
⎛ ⎞± −⎜ ⎟= < ≤⎜ ⎟⎝ ⎠⎛ ⎞+⎜ ⎟= + ≠⎜ ⎟⎝ ⎠
Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
IDENTIDADES DE FUNCS HIP 2 2senh 1x x
( )( )( )
2 2
2
cosh1 tgh sechctgh 1 cschsenh senh
cosh cosh
tgh tgh
x xx x
x x
x x
x x
− =
− =
− = −
− =
− = −
− =
( )( )
( )
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
=+
cosh senhcosh senh
x
x
e x xe x x−
= +
= − OTRAS
( ) ( )
2
2
2
0
4 2
4 discriminanteexp cos sen 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
1ln
loglog
loglog 0, 1
ln
ln
aa
u u
u u
v v v
d du dx duudx 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
−
= = ⋅
= ⋅
= ⋅ >
= ⋅
= ⋅
= + ⋅ ⋅
a ≠
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
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
=
= −
=
= −
=
= −
=
DERIV DE FUNCS TRIGO INVER
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
1sen1
1cos1
1tg1
1ctg1
si 11secsi 11si 11cscsi 11
1vers2
d duudx dxud duudx dxud duudx u dxd duudx u dx
ud duuudx dxu uud duuudx dxu u
d duudx dxu u
∠ = ⋅−
∠ = − ⋅−
∠ = ⋅+
∠ = − ⋅+
+ >⎧∠ = ± ⋅ ⎨− < −− ⎩
− >⎧∠ = ⋅ ⎨+ < −− ⎩
∠ = ⋅−
∓
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
senh cosh
cosh senh
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
{ }( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )( ) [ ]
( ) ( )( ) ( ) [ ]
( ) ( )
0
, , ,
,
si
a a ab b
a ab c b
a a cb a
a ba
ab
a
b b
a a
b b
a a
b b bf 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 u u
2
2
sen cos
cos sen
sec tg
csc ctg
sec tg sec
csc ctg csc
ud
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫∫∫∫∫∫
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 TRIGO INV
( )
( )
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
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 coshu=
2
2
senh
cosh senh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu
udu u
udu u
udu u
u udu u
u udu u
=
=
= −
= −
= −
∫∫∫∫∫∫
( )( )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
−
=
=
= ∠
= −
=
∫∫∫∫
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
sen
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
sen cossen
cos sencos
au
auau
e a bu b bue bu du
a be a bu b bu
e bu dua b
−=
++
=+
∫
∫
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! !
sen 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
−−
−
−−
+ + −−
+ = − + − + + −
∠ = − + − + + −−
Formulario de Cálculo Diferencial e Integral Jesús Rubí M. ÁLGEBRA LINEAL
Def. El determinante de una matriz
11 12
21 22
a aA
a a⎡ ⎤
= ⎢ ⎥⎣ ⎦
está dado por
11 1211 22 12 21
21 22
det .a a
A a a a aa a
= = −
Def. El determinante de una matriz
11 12 13
21 22 23
31 32 33
a a aA a a a
a a a
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
está dado por
11 22 33 12 23 3111 12 13
13 21 32 11 23 3221 22 23
31 32 33 12 21 33 13 22 31
det .a a a a a aa a aa a a a a aA a a a
a a a a a a a a a
⋅ ⋅ + ⋅ ⋅+ ⋅ ⋅ − ⋅ ⋅= =− ⋅ ⋅ − ⋅ ⋅
