Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
Formulario de
Cálculo Diferencial
e Integral
V
ER
.4.3
Jesús Rubí Miranda
(jesusrubim@yahoo.com)
http://mx.geocities.com/estadisticapapers/
http://mx.geocities.com/dicalculus/
VALOR ABSOLUTO
11
11
si 0
si 0
y
0 y 0 0
ó
ó
nn
kk
kk
nn
kk
kk
aa
aaa
aa
aa aa
aa a
ab a b a a
ab a b a a
==
==
≥
⎧
=⎨−<
⎩
=−
≤−≤
≥=⇔=
==
+≤ + ≤
∏∏
∑∑
EXPONENTES
()
()
/
pq pq
ppq
q
q
ppq
p
p
p
pp
p
q
pq p
aa a
aa
a
aa
ab a b
aa
bb
aa
+
−
⋅=
=
=
⋅=⋅
⎛⎞
=
⎜⎟
⎝⎠
=
LOGARITMOS
10
log
log log log
log log log
log log
log ln
log log ln
log log y log ln
x
a
aaa
aaa
r
aa
b
a
b
e
Nx a
MN M N
MMN
N
Nr N
NN
Naa
N
N
NNN
=⇒
=+
=−
=
==
==
=
ALGUNOS PRODUCTOS
ac ad
= +
()
()()
()()()
()()()
()() ()
()() ( )
()()
()
()
()
22
222
222
2
2
332 23
332 23
2222
2
2
33
33
222
acd
ab ab a b
ab ab ab a abb
ab ab ab a abb
xb xd x bdxbd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
ab a ab ab b
ab a ab ab b
a b c a b c ab ac bc
⋅+
+⋅−= −
+⋅+= + = + +
−⋅−= − = − +
+⋅+ = ++ +
+⋅ + = + + +
+⋅+ = + + +
+=+ + +
−=− + −
++ = + + + + +
1
1
nnk k n n
k
ab a abb a b
ab a abab b a b
ab a abab ab b a b
ab ab a b n
−−
=
−⋅ + + = −
−⋅ + + + = −
−⋅ + + + + = −
⎛⎞
−⋅ = − ∀∈
⎜⎟
⎝⎠
∑
()
()
()
()
()
()
()
2233
32 23 44
43 22 34 55
()
(
)
()
()
()
()
()
()
2233
32 23 44
43 22 34 55
54 3223 45 66
ab a abb a b
ab a abab b a b
ab a abab ab b a b
abaababababb ab
+⋅ −+ = +
+⋅ − + − = −
+⋅ − + − + =+
+⋅ − + − + − = −
()()
()()
11
1
11
1
1 impar
1 par
nknk k n n
k
nknk k n n
k
ab a b a b n
ab a b a b n
+−−
=
+−−
=
⎛⎞
+⋅ − = + ∀∈
⎜⎟
⎝⎠
⎛⎞
+⋅ − = − ∀∈
⎜⎟
⎝⎠
∑
∑
SUMAS Y PRODUCTOS
n
()
()
12 1
1
11
111
10
nk
k
n
k
nn
kk
kk
nnn
kk k k
kkk
n
kk n
k
aa a a
cnc
ca c a
ab a b
aa aa
=
=
==
===
−
=
+++=
=
=
+= +
−=−
∑
∑
∑∑
∑∑∑
∑
()
1
()
()
()
()
()
()
()
1
1
1
2
1
232
1
3432
1
4 543
1
2
1
121
2
= 2
1
11
1
2
123
6
12
4
1615 10
30
135 2 1
!
n
k
n
nk
k
n
k
n
k
n
k
n
k
n
k
n
ak d an d
nal
rarl
ar a rr
knn
knnn
knnn
knnnn
nn
nk
nn
k
=
−
=
=
=
=
=
=
+− = +−⎡⎤⎡ ⎤
⎣⎦⎣ ⎦
+
−−
==
−−
=+
=++
=++
=++−
+++ + − =
=
⎛⎞
=
⎜⎟
⎝⎠
∑
∑
∑
∑
∑
∑
∏
()
()
0
!,
!!
n
nnk k
k
kn
nkk
n
xy xy
k
−
=
≤
−
⎛⎞
+=
⎜⎟
⎝⎠
∑
()
12
12 12
12
!
!! ! k
nnnn
kk
k
n
x
xx xxx
nn n
+++ = ⋅
∑
CONSTANTES
9…3.1415926535
2.71828182846e
π
=
=…
TRIGONOMETRÍA
1
sen csc sen
1
cos sec cos
sen 1
tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θθ
θ
θθ
θ
θ
θθ
θ
θ
==
==
== =
radianes=180
π
CA
CO
H
I
P
θ
θ
sen cos tg ctg sec csc
0010
∞
1
∞
3012
3
2 1
3
32
3
2
451212 1 1 2 2
6032 12 313 223
9010
∞
0
∞
1
[]
[]
sen ,
22
cos 0,
tg ,
22
1
ctg tg 0,
1
sec cos 0,
1
csc sen ,
22
yxy
yxy
yxy
yx y
x
yx y
x
yx y
x
ππ
π
ππ
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
2
cos 1
θθ
2
22
22
sen
1ctg csc
tg 1 sec
θ
θ
θ
θ
+
=
+=
+=
()
()
()
sen sen
cos cos
tg tg
θ
θ
θ
θ
θθ
−=−
−=
−=−
()
()
()
()
()
()
()()
()()
()
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
θπ θ
θπ θ
θπ θ
θπ θ
θπ θ
θπ θ
θ
πθ
θ
πθ
θπ θ
+=
+=
+=
+=−
+=−
+=
+=−
+=−
+=
()
()()
()
()
sen 0
cos 1
tg 0
21
sen 1
2
21
cos 0
2
21
tg 2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
=−
=
+
⎛⎞
=−
⎜⎟
⎝⎠
+
⎛⎞
=
⎜⎟
⎝⎠
+
⎛⎞
=∞
⎜⎟
⎝⎠
sen cos 2
cos sen 2
π
θθ
π
θθ
⎛⎞
=−
⎜⎟
⎝⎠
⎛⎞
=+
⎜⎟
⎝⎠
()
()
()
()
()
22
2
2
2
2
sen sen cos cos sen
cos cos cos sen sen
tg tg
tg 1tgtg
sen2 2 sen cos
cos2 cos sen
2tg
tg2 1tg
1
sen 1 cos 2
2
1
cos 1 cos 2
2
1cos2
tg 1cos2
α
βαβαβ
α
βαβαβ
αβ
αβ αβ
θθθ
θθθ
θ
θθ
θθ
θθ
θ
θθ
±= ±
±=
±
±=
=
=−
=−
=−
=+
−
=+
∓
∓
() ()
() ()
() ()
() ()
11
sen sen 2sen cos
22
11
sen sen 2sen cos
22
11
cos cos 2cos cos
22
11
cos cos 2sen sen
22
α
βαβαβ
αβ αβ αβ
αβ αβ αβ
α
βαβαβ
+= +⋅ −
−= −⋅ +
+= +⋅ −
−=− +⋅ −
()
sen
tg tg cos cos
α
β
αβ
α
β
±
±= ⋅
()()
()()
()()
1
sen cos sen sen
2
1
sen sen cos cos
2
1
cos cos cos cos
2
α
βαβαβ
α
βαβαβ
α
βαβαβ
⋅= −+ +
⎡
⎤
⎣
⎦
⋅= −− +
⎡
⎤
⎣
⎦
⋅= −+ +
⎡
⎤
⎣
⎦
tg tg
tg tg ctg ctg
αβ
αβ
αβ
+
⋅= +
FUNCIONES HIPERBÓLICAS
senh 2
cosh 2
senh
tgh cosh
1
ctgh tgh
12
sech cosh
12
csch senh
xx
xx
x
x
xx
xx
xx
x
ee
x
xee
xxee
ee
xxee
x
xx
ee
x
x
x
ee
x
x
ee
−
−
−
−
−
−
−
=
+
=
−
==
+
+
==
−
==
+
==
−
−
−
[
{}
]
{} {}
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
(
)
()
12
12
1
1
2
1
2
1
senh ln 1 ,
cosh ln 1 , 1
11
tgh ln , 1
21
11
ctgh ln , 1
21
11
sech ln , 0 1
11
csch ln , 0
xxx x
xxx x
x
xx
x
x
xx
x
x
xx
x
x
xx
xx
−
−
−
−
−
−
=++∀∈
=±− ≥
+
⎛⎞
=<
⎜⎟
−
⎝⎠
+
⎛⎞
=>
⎜⎟
−
⎝⎠
⎛⎞
±−
⎜⎟
=
<≤
⎜⎟
⎝⎠
⎛⎞
+
⎜⎟
=+ ≠
⎜⎟
⎝⎠
