¡Descarga calculo para todos y más Apuntes en PDF de Informática solo en Docsity!
Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3)
http://www.geocities.com/calculusjrm/
Jesús Rubí M.
Fórmulas de Cálculo Diferencial e Integral
V
ER
Jesús Rubí Miranda
http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO^1
1 1
1
si
si
y 0 y
ó
n^ ó
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
q p^
pq p^
p^
p
p^
p p q
p q
p
a^
a^
a
a^
a
a a
a
a b
a^
b
a^
a
b^
b
a^
a
−
⋅^
⋅^
=^
⎛^
⎞^
⎜^
⎝^
LOGARITMOS
10 loglog
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
NN^
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
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
n^^1
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
impar
par
n^
k^
n^ k^
k^
n^
n
kn^
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 kn^
n
k^
k
k^
k
n^
n^
n
k^
k^
k^
k
k^
k^
k
n
k^
k^
n
a^ k
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
n k!
n
n^
k
kn kn kn kn 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
π = e
=^
TRIGONOMETRÍA
sen
csc
sen
cos
sec
cos
sen
tg
ctg
cos
tg
COHIPCAHIP
COCA
θ
θ^
θ
θ
θ^
θ
θ
θ
θ
θ
θ
=^
=^
=^
=^
radianes=
D
CA
CO
H IP θ
θ^
sin
cos
tg
ctg
sec
csc
D 0
D^
D^
D
D^
[^
]
[^
]
sin
cos
tg
ctg
tg
sec
cos
csc
sen
y^
x^
y
y^
x^
y
y^
x^
y
y^
x^
y x
y^
x^
y x
y^
x^
y π^ x
π π π^
π
π π π^
π
⎡^
⎣^
⎡^
−⎢^
⎣^
Gráfica 1. Las funciones trigonométricas:
sin
x
cos
x^
,^
tg
x^
-^ -^ -^ -^
0
2
4
6
8
(^2) 1.5 (^1) 0.5 (^0) -0.5 -1 -1.5 -
sen xcos xtg x
Gráfica 2. Las funciones trigonométricas
csc
x
sec
x
ctg
x
-^ -^ -^ -^
0
2
4
6
8
2.5^2 1.5^1 0.5^0 -0.5^ -1 -1.5^ -2 -2.
csc xsec xctg x
Gráfica 3. Las funciones trigonométricas inversasarcsin
x
,^
arccos
x
,^
arctg
x
-^ -^ -^
0
1
2
3
(^43210) -1 -
arc sen xarc cos xarc tg x
Gráfica 4. Las funciones trigonométricas inversasarcctg
x^
,^
arcsec
x
arccsc
x
-^
0
5
(^43210) -1 -
arc ctg xarc sec xarc csc x
IDENTIDADES TRIGONOMÉTRICAS 2
2
2
2
2
2
sin
cos
ctg
csc
tg
sec
θ^
θ
θ^
θ
θ
θ
+^
+^
(^
(^
(^
sin
sin
cos
cos
tg
tg
θ
θ
θ^
θ
θ
θ
−^
−^
−^
(^
(^
(^
(^
(^
(^
(^
)^
(^
(^
)^
(^
(^
sin
sin
cos
cos
tg
tg
sin
sin
cos
cos
tg
tg
sin
sin
cos
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
cos
tg
sin
cos
tg
n
n
n n n π π π n n n
π π π = =
⎛^
⎜^
⎝^
⎛^
⎜^
⎝^
⎛^
⎜^
⎝^
sin
cos
cos
sin
π 2
θ
θ
π
θ^
⎛^ θ
=^
⎜^
⎝^
⎛^
=^
⎜^
⎝^
(^
(^
(^
)^ (
(^
2
2
2
(^222) sin
sin
cos
cos
sin
cos
cos
cos
sin
sin
tg
tg
tg
tg
tg
sin 2
2sin
cos
cos 2
cos
sin
2 tg
tg 2
tg 1
sin
cos 2
21
cos
cos 2
2 1
cos 2
tg
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
=^
±^
±^
=^
=^
(^
)^
(^
(^
)^
(^
(^
)^
(^
(^
)^
(^
sin
sin
2sin
cos
sin
sin
2 sin
cos
cos
cos
2 cos
cos
cos
cos
2 sin
sin
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 ±
±^
=^
(^
)^
(^
(^
)^
(^
(^
)^
(^
sin
cos
sin
sin
sin
sin
cos
cos
cos
cos
cos
cos
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
cosh
sinh
tgh
cosh
ctgh
tgh
sech
cosh
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 :
tgh :
ctgh :
,^
sech :
csch :
−^
→−^
\
\
\ \ \ \ \
\
Gráfica 5. Las funciones hiperbólicas
sinh
x
cosh
x
tgh
x
-^
0
5
(^543210) -1 -2 -3 -
senh xcosh xtgh x
FUNCIONES HIPERBÓLICAS INV
(^
(^
1
2
1
2
1 1
2
1
2
1
sinh
ln
cosh
ln
tgh
ln
,^
ctgh
ln
,^
sech
ln
csch
ln
,^
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
− − − − − −
=^
+^
+^
=^
±^
−^
⎛^
=^
⎜^
⎝^
⎛^
⎜^
⎝^
⎛^
±^
⎜^
=^
<^
⎜^
⎝^
⎛^
⎜^
=^
⎜^
⎝^
\
Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3)
http://www.geocities.com/calculusjrm/
Jesús Rubí M.
IDENTIDADES DE FUNCS HIP
2
2 sinh
x^
x
(^
(^
(^
2
2
2
cosh 1
tgh
sech
ctgh
csch
sinh
sinh
cosh
cosh
tgh
tgh x^
x
x^
x
x^
x
x^
x
x^
x
−^
−^
−^ = −
(^
(^
(^
2
2
2
sinh
sinh
cosh
cosh
sinh
cosh
cosh
cosh
sinh
sinh
tgh
tgh
tgh
tgh
tgh
sinh 2
2sinh
cosh
cosh 2
cosh
sinh
2 tgh
tgh 2
tgh
x^
y^
x^
y^
x^
y
x^
y^
x^
y^
x^
y
x^
y
x^
y^
x^
y
x^
x^
x
x^
x^
x
x
x^
x
±^
=^
±^
=^
±^
=^
=^
+^ (
(^
(^222)
sinh
cosh 2
cosh
cosh 2
(^2) cosh 2
tgh
cosh 2
x^
x
x^
x x
x^
x
=^
=^
=^
sinh 2
tgh
cosh 2
x
x^
x
=^
cosh
sinh
cosh
sinh
xe x
x^
x
e^
x^
x
−
=^
OTRAS
(^
)^
(^
2
2
2
discriminante
exp
cos
sin
si
ax
bx
c
b^
b^
ac
x^
a
b^
aci
e^
i
α
α^
β
β^
β
α β
+^
= −^
±^
\
LÍMITES
(^
0 0 0 0 1 lim 1
lim 1
sen lim
cos
lim
lim
lim
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
lim 1
lim
x^
x^
x
n^
n
f^
x^
x^
f^
x
df
y
D f
x^
dx
x^
x
d^
c dxd
cx
c
dxd
cx
ncx
dxd
du
dv
dw
u^
v^
w
dx
dx
dx
dx
d^
du
cu
c
dx
dx
∆ →
∆ →
−
−^
=^
=^
±^
±^
±^
=^
±^
±^
(^
(^
(^
)^
(^
(^
2
1
n^
n
d^
dv
du
uv
u^
v
dx
dx
dx
d^
dw
dv
du
uvw
uv
uw
vw
dx
dx
dx
dx
v du dx
u dv dx
d^
u dx
v^
v
d^
du
u^
nu
dx
dx −
=^
=^
+^
⎛^
⎜^
⎝^
2 1
2
(Regla de la Cadena)
donde
dF
dF
du
dx
du
dx
dudx
dx dudF du
dFdx
dx du
x^
f^
t
f^
t
dy dt
dydx
dx dt
f^
t^
y^
f^
t
′^
=^
=^
′^
DERIVADA DE FUNCS LOG & EXP
(^
(^
(^
(^
(^
(^
)^
1
ln
log
log
log
log
ln
ln a
a u^
u
u^
u
v^
v^
v
u
dx
u^
u^
dx
d^
e^
du
u
dx
u^
dxe
d^
du
u^
a
dx
u^
dx
d^
du
e^
e
dx
dx
d^
du
a^
a^
a
dx
dx
d^
du
dv
u^
vu
u u
dx
dx
dx
−
=^
=^
⋅^
=^
=^
+^
⋅^
d^
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^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u^
u
dx
dx
d^
du
u^
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
DERIV DE FUNCS TRIGO INVER
(^
(^
(^
(^
(^
(^
(^
2
2 2
2 2 2
2
sin
cos
tg
ctg
si
sec
si
si
csc
si
vers
u
dx
dx u
d^
du
u
dx
dx u
d^
du
u
dx
dx u
d^
du
u
dx
dx u
u
d^
du
u^
u
dx
dx
u^
u
u
d^
du
u^
u
dx
dx
u^
u
d^
du
u
dx
dx
u^
u
+^
⋅^
d^
du
−^
=^
⋅^
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
sinh
cosh
cosh
sinh
tgh
sech
ctgh
csch
sech
sech
tgh
csch
csch
ctgh
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u
dx
dx
d^
du
u^
u^
u
dx
dx
d^
du
u^
u^
u
dx
dx
d^
du
DERIVADA DE FUNCS HIP INV^1
2
1
2
1
2
1
2
1
1
1
2
senh
si cosh
cosh
,^
si cosh
tgh
,^
ctgh
,^
si sech
sech
si sech
u
dx
dx u
u
d^
du
u^
u
dx
dx
u
u
d^
du
u^
u
dx
u^
dx
d^
du
u^
u
dx
u^
dx
u^
u
d^
du
u
dx
dx
u^
u
u^
u
− − − −
−
−
−
⋅^
⋅^
=^
⋅^
=^
⋅^
d^
du
1
2 1
csch
,^
d^
du
u^
u
dx
dx
u^
u
−
⋅^
INTEGRALES DEFINIDAS, PROPIEDADES
Nota. Para todas
las fórmulas de integración deberá
agregarse una constante arbitraria
c^
(constante de
integración).
(^
)^
(^
{^
}^
(^
)^
(^
(^
)^
(^
(^
)^
(^
)^
(^
(^
)^
(^
(^
(^
)^
(^
)^
(^
(^
)^
[^
]
(^
)^
(^
(^
)^
(^
)^
[^
]
(^
)^
(^
,^
,^
, 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
ln
n
adx n
ax
af
x dx
a^
f^
x dx
u^
v^
w
dx
udx
vdx
wdx
udv
uv
vdu u
u du
n
n
du
u
u
=
±^
±^
±^
±^
=^
=^
∫ ∫^
∫^
∫^
∫^
∫^
INTEGRALES DE FUNCS LOG & EXP
(^
(^
(^
)^
(^
(^
(^
2 2
ln
ln
ln 1
ln
ln
ln
log
ln
ln
ln
ln
log
2log
ln
2ln
u^
u u
u
u
u u^
u
a a^
a
e du
e
a a
a du
a a a
ua du
u a^
a
ue du
e^
u
udu
u^
u^
u^
u^
u
u
udu
u^
u^
u^
u
a^
a
u
u^
udu
u
u
u^
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
sin
sin 2
2
cos
sin 2
2
tg
tg
ctg
ctg u
udu
u
u
udu
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
cos
cos
tg
tg
ln
ctg
ctg
ln
sec
sec
ln
sec
cosh
csc
csc
ln
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 tgh
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 21
ln 2
du u^
a^
a^
au
a^
a
du
u^
a^
u^
a
u^
a^
a^
u^
a
du
a^
u^
u^
a
a^
u^
a^
a^
u
=^
=^
−^
=^
−^
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
sin cos ln
ln 1
cos 1
sec
sen
ln
du
ua
a^
u
ua
du
u^
u^
a
u^
a du
u
a
u^
a^
u^
a^
a^
u
du
a
a^
u
u^
u^
a
u
a^
a
u^
a^
u
a^
u du
a^
u^
a
u^
a
u^
a du
u^
a^
u^
u^
a
= −∠=^
+^
=^
−^
=^
−^
+^
±^
=^
±^
±^
+^
MÁS INTEGRALES
(^
(^
2
2 2
2
3
sin
cos
sin
cos
sin
cos
sec
sec
tg
ln sec
tg
au
au
au
e^
a^
bu
b^
bu
e^
bu du
a^
b
e^
a^
bu
b^
bu
e^
bu du
a^
b
u du
u^
u^
u^
u
=^
=^
=^
au ALGUNAS SERIES
(^
)^
(^
)^
(^
)^
(^
(^ )
(^
)^
(^
)^
(^
)^
(^
(^ )
(^
)^
(^
2
0
0
0
0
0
0
0
2
2
3
3
5
7
2
1
1
2
4
6
'^
: Taylor
: Maclaurin
!
sin
cos
n
n n^
n
n
x
n
f^ n
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^
x
e^
x^
n
x^
x^
x^
x
x^
x^
n
x^
x^
x
x
−
−
−^
+^
+^
+^
=^
+^
+^
+^
=^
−^
+^
+^
=^
+^
(^
)^
(^
(^
)^
(^
(^
2
2
1
2
3
4
1
3
5
7
2
1
1 1
ln 1
tg
n
n
n n
n
n
x n
x^
x^
x^
x
x^
x^
n
x^
x^
x^
x
x^
x^
n
−
−
−
−
−
+^
+^
=^
−^
+^
+^
=^
−^
+^
+^
+^
