Formulario De Calculo Diferencial E Integral

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Formulario de Cálculo Diferencial e Integral (Página 1 de 2) Jesús Rubí M. 
Formulario de 
Cálculo Diferencial 
e Integral VER.4.9 
Jesús Rubí Miranda (jesusrubim@yahoo.com) 
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 a b a a
a b a b a a
= =
= =
≥⎧= ⎨− <⎩
= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
 
EXPONENTES 
( )
( )
/
p q p q
p
p q
q
qp pq
p p p
p p
p
qp q p
a a a
a a
a
a a
a b a b
a a
b b
a a
+
−
⋅ =
=
=
⋅ = ⋅
⎛ ⎞ =⎜ ⎟⎝ ⎠
=
 
LOGARITMOS 
10
log
log log log
log log log
log log
log lnlog
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 NN
a 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
 
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
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
nn
k
k
n
k
n
k
n
k
n
k
n
k
na k d a n d
n a l
r a rlar 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 21 2 1 2
1 2
!
! ! !
k
n nn n
k k
k
nx x x x x x 
n n n
+ + + = ⋅∑" ""
CONSTANTES 
9…3.1415926535
2.71828182846e
π =
= … 
TRIGONOMETRÍA 
1sen csc
sen
1cos sec
cos
sen 1tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θ θ θ
θ θ θ
θθ θθ θ
= =
= =
= = =
 
 radianes=180π D 
 
CA 
CO
HIP 
θ 
 
 
θ sen cos tg ctg sec csc
0D 0 1 0 ∞ 1 ∞
30D 1 2 3 2 1 3 3 2 3 2
45D 1 2 1 2 1 1 2 2
60D 3 2 1 2 3 1 3 2 2 3
90D 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 y
x
y x y
x
y x y
x
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 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 
arcsen 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 5
-2
-1
0
1
2
3
4
arc ctg x
arc sec x
arc csc x
 
IDENTIDADES TRIGONOMÉTRICAS 
2cos 1θ θ2
2 2
2 2
sen
1 ctg csc
tg 1 sec
θ θ
θ 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 1
2
2 1cos 0
2
2 1tg
2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+⎛ ⎞ = −⎜ ⎟⎝ ⎠
+⎛ ⎞ =⎜ ⎟⎝ ⎠
+⎛ ⎞ = ∞⎜ ⎟⎝ ⎠
 
sen cos
2
cos sen
2
πθ θ
πθ θ
⎛ ⎞= −⎜ ⎟⎝ ⎠
⎛ ⎞= +⎜ ⎟⎝ ⎠
 
( )
( )
( )
( )
( )
2 2
2
2
2
2
sen sen cos cos sen
cos cos cos sen sen
tg tgtg
1 tg tg
sen 2 2sen cos
cos 2 cos sen
2 tgtg 2
1 tg
1sen 1 cos 2
2
1cos 1 cos 2
2
1 cos 2tg
1 cos 2
α β α β α β
α β α β α β
α βα β α β
θ θ θ
θ θ θ
θθ θ
θ θ
θ θ
θθ θ
± = ±
± =
±± =
=
= −
= −
= −
= +
−= +
∓
∓
 
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1sen sen 2sen cos
2 2
1 1sen sen 2sen cos
2 2
1 1cos cos 2cos cos
2 2
1 1cos cos 2sen sen
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
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
 
( )sen
tg tg
cos cos
α βα β α β
±± = ⋅ 
( ) ( )
( ) ( )
( ) ( )
1sen cos sen sen
2
1sen sen cos cos
2
1cos cos cos cos
2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
 
tg tgtg tg
ctg ctg
α βα β α β
+⋅ = + 
FUNCIONES HIPERBÓLICAS 
senh
2
cosh
2
senhtgh
cosh
1ctgh
tgh
1 2sech
cosh
1 2csch
senh
x x
x x
x
x
x x
x x
x x
x
e ex
x e ex
x e e
e ex
x e e
x
x xe e
x
x
x e e
x
x 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 x
cosh x
tgh x
 
FUNCIONES HIPERBÓLICAS INV 
( )
( )
1 2
1 2
1
1
2
1
2
1
senh ln 1 , 
cosh ln 1 , 1
1 1tgh ln , 1
2 1
1 1ctgh ln , 1
2 1
1 1sech ln , 0 1
1 1csch ln , 0
x x x x
x x x x
xx x
x
xx x
x
xx x
x
xx x
x x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞= <⎜ ⎟−⎝ ⎠
+⎛ ⎞= >⎜ ⎟−⎝ ⎠
⎛ ⎞± −⎜ ⎟= < ≤⎜ ⎟⎝ ⎠
⎛ ⎞+⎜ ⎟= + ≠⎜ ⎟⎝ ⎠
\
 
 
Formulario de Cálculo Diferencial e Integral (Página 2 de 2) Jesús Rubí M. 
 
IDENTIDADES DE FUNCS HIP2 2senh 1x x
( )
( )
( )
2 2
2
cosh
1 tgh sech
ctgh 1 csch
senh senh
cosh cosh
tgh tgh
x x
x x
x x
x x
x x
− =
− =
− = −
− =
− = −
− =
 
( )
( )
( )
2 2
2
senh senh cosh cosh senh
cosh cosh cosh senh senh
tgh tghtgh
1 tgh tgh
senh 2 2senh cosh
cosh 2 cosh senh
2 tghtgh 2
1 tgh
x y x y x y
x y x y x y
x yx y
x y
x x x
x x x
xx
x
± = ±
± = ±
±± = ±
=
= +
= +
 
( )
( )
2
2
2
1senh cosh 2 1
2
1cosh cosh 2 1
2
cosh 2 1tgh
cosh 2 1
x x
x x
xx
x
= −
= +
−= +
 
senh 2tgh
cosh 2 1
xx
x
= + 
cosh senh
cosh senh
x
x
e x x
e x x−
= +
= − 
OTRAS 
( ) ( )
2
2
2
0
4 
2
 4 discriminante
exp cos sen si ,
ax bx c
b b acx
a
b ac
i 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 1
ln
x
x
x
x
x
x
x
x
x
x e
e
x
x
x
x
x
e
x
x
x
→
→∞
→
→
→
→
+ = =
⎛ ⎞+ =⎜ ⎟⎝ ⎠
=
− =
− =
− =
 
DERIVADAS 
( ) ( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x x x
n n
f x x f xdf yD f x
dx x x
d c
dx
d cx c
dx
d cx ncx
dx
d du dv dwu v w
dx dx dx dx
d ducu c
dx dx
∆ → ∆ →
−
+ ∆ − ∆= = =∆ ∆
=
=
=
± ± ± = ± ± ±
=
" "
 
( )
( )
( ) ( )
( )
2
1n n
d dv duuv u v
dx dx dx
d dw dv duuvw uv uw vw
dx dx dx dx
v du dx u dv dxd u
dx v v
d duu nu
dx dx
−
= +
= + +
−⎛ ⎞ =⎜ ⎟⎝ ⎠
=
 
( )
( )
( )
( )
12
1 2
 (Regla de la Cadena)
1
 donde
dF dF du
dx du dx
du
dx dx du
dF dudF
dx dx du
x f tf tdy dtdy
dx dx dt f t y f t
= ⋅
=
=
=⎧′ ⎪= = ⎨′ =⎪⎩
 
DERIVADA DE FUNCS LOG & EXP 
( )
( )
( )
( )
( )
( ) 1
1ln
loglog
loglog 0, 1
ln
ln
a
a
u u
u u
v v v
d du dx duu
dx u u dx
d e duu
dx u dx
ed duu a
dx u dx
d due e
dx dx
d dua a a
dx dx
d du dvu vu u u
dx 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 u
dx dx
d duu u
dx dx
d duu u
dx dx
d duu u
dx dx
d duu u u
dx dx
d duu u u
dx dx
d duu u
dx dx
=
= −
=
= −
=
= −
=
 
DERIV DE FUNCS TRIGO INVER 
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
1sen
1
1cos
1
1tg
1
1ctg
1
si 11sec
si 11
si 11csc
si 11
1vers
2
d duu
dx dxu
d duu
dx dxu
d duu
dx u dx
d duu
dx u dx
ud duu
udx dxu u
ud duu
udx dxu u
d duu
dx dxu u
∠ = ⋅−
∠ = − ⋅−
∠ = ⋅+
∠ = − ⋅+
+ >⎧∠ = ± ⋅ ⎨− < −− ⎩
− >⎧∠ = ⋅ ⎨+ < −− ⎩
∠ = ⋅−
∓
 
DERIVADA DE FUNCS HIPERBÓLICAS 
2
2
senh cosh
cosh senh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
u u
dx dx
d duu u
dx dx
d duu u
dx dx
d duu u
dx dx
d duu u u
dx dx
d duu u u
dx dx
=
=
=
= −
= −
= −
d du
 
DERIVADA DE FUNCS HIP INV 
1
2
-1
1
-12
1
2
1
2
1
1
12
senh
1
 si cosh 01cosh , 1 
 si cosh 01
1tgh , 1
1
1ctgh , 1
1
si sech 0, 0,11sech
si sech 0, 0,11
u
dx dxu
ud duu u
dx dx uu
d duu u
dx u dx
d duu u
dx u dx
u ud duu
dx dx u uu u
−
−
−
−
−
−
−
= ⋅+
⎧+ >± ⎪= ⋅ > ⎨− <− ⎪⎩
= ⋅ <−
= ⋅ >−
⎧− > ∈⎪= ⋅ ⎨+ < ∈− ⎩
∓
1d du
1
2
1csch , 0
1
d duu u
dx 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 a
b b
a a
b c b
a a c
b a
a b
a
a
b
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
 1
1
ln
n
n
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
uu du n
n
du u
u
+
=
=
± ± ± = ± ± ±
= −
= ≠ −+
=
∫
∫ ∫
∫ ∫ ∫ ∫
∫ ∫
∫
∫
" "
 
INTEGRALES DE FUNCS LOG & EXP 
( )
( )
( ) ( )
( )
( )
2
2
0
1ln
1
ln ln
1
ln ln ln 1
1log ln ln 1
ln ln
log 2log 1
4
ln 2ln 1
4
u u
u
u
u
u
u u
a
a a
e du e
aaa du
aa
aua du u
a a
ue du e u
udu u u u u u
uudu u u u u
a a
uu 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 2
2 4
1cos sen 2
2 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 tgh
2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫
∫
∫
∫
 
INTEGRALES DE FRAC 
( )
( )
2 2
2 2
2 2
2 2
2 2
tg
1 ctg
1 ln 
2
1 ln 
2
du
u a a a
u
a a
du u a u a
u a a u a
du a u u a
a u a a u
= ∠+
= − ∠
−= >− +
+= <− −
∫
∫
∫
1 u
 
INTEGRALES CON 
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
2
2 2 2 2
2
2 2 2 2 2 2
sen
 cos
ln
1 ln
1 cos
1 sec
sen
2 2
ln
2 2
du u
aa u
u
a
du u u a
u a
du u
au a u a a u
du a
a uu u a
u
a a
u a ua u du a u
a
u au a du u a u u a
= ∠−
= −∠
= + ±±
=± + ±
= ∠−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
 
MÁS INTEGRALES 
( )
( )
2 2
2 2
3
sen 
cos sen
cos 
1 1sec sec tg ln sec tg
2 2
au
au
au
u
e bu du
a b
e a bu b bu
e bu du
a b
u du u u u u
= +
+= +
= + +
∫
∫
∫
sen cosaue a bu b b−
 
ALGUNAS SERIES 
( ) ( ) ( )( ) ( )( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
2
0 0
0 0 0
0 0
2
2 3
3 5 7 2 1
1
2 4 6
'
2!
 : Taylor
!
'' 0
0 ' 0
2!
0
 : Maclaurin
!
1
2! 3! !
sen1
3! 5! 7! 2 1 !
cos 1
2! 4!
nn
n n
n
x
n
n
f x x x
f x f x f x x x
f x x x
n
f x
f x f f x
f x
n
x x xe x
n
x x x xx x
n
x x xx
−−
−= + − +
−+ +
= + +
+ +
= + + + + + +
= − + − + + − −
= − + −
"
"
" "
"
''
( ) ( )
( ) ( )
( )
2 2
1
2 3 4
1
3 5 7 2 1
1
1
6! 2 2 !
ln 1 1
2 3 4
tg 1
3 5 7 2 1
n
n
n
n
n
n
x
n
x x x xx x
n
x x x xx x
n
−−
−
−−
+ + − −
+ = − + − + + −
∠ = − + − + + − −
"
"
"