FORMULARIO DE
CÁLCULO DIFERENCIAL
VER.3.7
E INTEGRAL
Jesús Rubí Miranda (jesusrubi1@yahoo.com)
http://mx.geocities.com/estadisticapapers/
http://mx.geocities.com/dicalculus/
VALOR ABSOLUTO
 a si a ≥ 0
a =
 −a si a < 0
a = −a
n
= ∏ ak
k
k =1
n
n
∑a
k =1
≤ ∑ ak
k
k =1
EXPONENTES
a ⋅a = a
p
q
( a ⋅ b)
k =1
1
k =1
k =1
− ak −1 ) = an − a0
k
n
n
n
∑ ar
k −1
k =1
n
(a + l )
2
n
1− r
a − rl
=a
=
1− r
1− r
n
1
∑ k = 2 (n
q
ap/q = ap
k =1
LOGARITMOS
n
log a MN = log a M + log a N
M
= log a M − log a N
N
log a N r = r log a N
log a
logb N ln N
=
logb a
ln a
2
ALGUNOS PRODUCTOS
a ⋅ ( c + d ) = ac + ad
(a + b) ⋅ ( a − b) = a − b
2
( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2
2
−
⋅
−
=
−
a
b
a
b
a
b
(
)(
) (
) = a 2 − 2ab + b 2
( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd
( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd
( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd
3
( a + b ) = a3 + 3a 2 b + 3ab 2 + b3
3
( a − b ) = a 3 − 3a 2 b + 3ab 2 − b3
2
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
2
( a − b ) ⋅ ( a 2 + ab + b 2 ) = a 3 − b3
( a − b ) ⋅ ( a3 + a 2 b + ab2 + b3 ) = a 4 − b 4
( a − b ) ⋅ ( a 4 + a 3b + a 2 b 2 + ab3 + b 4 ) = a 5 − b5
 n

( a − b ) ⋅  ∑ a n −k b k −1  = a n − b n ∀n ∈
 k =1

0
∞
y = ∠ cos x
 π π
y ∈ − , 
 2 2
y ∈ [ 0, π ]
y = ∠ tg x
y∈ −
csc
1
∞
2
3
2
2
2
1
1
3
2
2
0
∞
1
cos ( −θ ) = cos θ
sen (θ + 2π ) = sen θ
cos (θ + 2π ) = cosθ
tg (θ + 2π ) = tg θ
sen (θ + π ) = − sen θ
cos (θ + π ) = − cosθ
tg (θ + π ) = tg θ
sen (θ + nπ ) = ( −1) sen θ
n
cos (θ + nπ ) = ( −1) cos θ
tg (θ + nπ ) = tg θ
0.5
-0.5
cos ( nπ ) = ( −1)
-1
-2
-8
-6
-4
-2
0
2
4
6
8
Gráfica 2. Las funciones trigonométricas csc x ,
n
sec x , ctg x :
k =1
n
n!
, k≤n
 =
 k  ( n − k )! k !
n
n


n
( x + y ) = ∑   x n−k y k
k =0  k 
n
2.5
2
1.5
CONSTANTES
π = 3.14159265359…
e = 2.71828182846…
TRIGONOMETRÍA
CO
sen θ =
HIP
CA
cosθ =
HIP
sen θ CO
=
tg θ =
cos θ CA
0
nk
k
x
-0.5
-1
csc x
sec x
ctg x
-2
-2.5
-8
1
sen θ
1
secθ =
cos θ
1
ctg θ =
tg θ
cscθ =
-6
-4
-2
0
2
4
6
8
Gráfica 3. Las funciones trigonométricas inversas
arcsen x , arccos x , arctg x :
4
3
2
1
0
-1
-2
-3
n
 2n + 1 
sen 
π  = ( −1)
 2

 2n + 1 
cos 
π=0
 2

 2n + 1 
tg 
π=∞
 2

sen (α ± β ) = sen α cos β ± cos α sen β
-1.5
arc sen x
arc cos x
arc tg x
-2
-1
0
1
2
3
tg α + tg β
ctg α + ctg β
e x − e− x
2
e x + e− x
cosh x =
2
senh x e x − e − x
tgh x =
=
cosh x e x + e− x
1
e x + e− x
=
ctgh x =
tgh x e x − e − x
1
2
=
sech x =
cosh x e x + e − x
1
2
csch x =
=
senh x e x − e − x
senh x =
cosh :
tgh :
ctgh :
→
→ [1, ∞
→ −1,1
− {0} → −∞ , −1 ∪ 1, ∞
sech :
→ 0,1]
csch :
− {0} →
cos (α ± β ) = cos α cos β ∓ sen α sen β
tg α ± tg β
tg (α ± β ) =
1 ∓ tg α tg β
sen 2θ = 2sen θ cosθ
cos 2θ = cos 2 θ − sen 2 θ
2 tg θ
tg 2θ =
1 − tg 2 θ
1
sen 2 θ = (1 − cos 2θ )
2
1
cos 2 θ = (1 + cos 2θ )
2
1 − cos 2θ
tg 2 θ =
1 + cos 2θ
− {0}
Gráfica 5. Las funciones hiperbólicas senh x ,
cosh x , tgh x :
5
π

sen θ = cos  θ − 
2

π

cos θ = sen θ + 
2

1
0.5
n!
=∑
x1n1 ⋅ x2n2
n1 !n2 ! nk !
n
tg ( nπ ) = 0
sen x
cos x
tg x
-1.5
1
sen (α − β ) + sen (α + β ) 
2
1
sen α ⋅ sen β = cos (α − β ) − cos (α + β ) 
2
1
cos α ⋅ cos β = cos (α − β ) + cos (α + β ) 
2
sen α ⋅ cos β =
senh :
sen ( nπ ) = 0
0
sen (α ± β )
cos α ⋅ cos β
FUNCIONES HIPERBÓLICAS
tg ( −θ ) = − tg θ
y ∈ 0, π
5
tg α ⋅ tg β =
n
n! = ∏ k
π radianes=180
2
sen ( −θ ) = − sen θ
1
+ ( 2n − 1) = n 2
+ xk )
IDENTIDADES TRIGONOMÉTRICAS
sen θ + cos 2 θ = 1
tg 2 θ + 1 = sec 2 θ
1.5
=
( x1 + x2 +
0
1 + ctg 2 θ = csc2 θ
2
+ n)
tg α ± tg β =
arc ctg x
arc sec x
arc csc x
-2
-5
3
π π
,
2 2
1
x
1
1
(α + β ) ⋅ cos (α − β )
2
2
1
1
sen α − sen β = 2sen (α − β ) ⋅ cos (α + β )
2
2
1
1
cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β )
2
2
1
1
cos α − cos β = −2sen (α + β ) ⋅ sen (α − β )
2
2
sen α + sen β = 2 sen
0
-1
k =1
1+ 3 + 5 +
log10 N = log N y log e N = ln N
2
1
( 2n3 + 3n2 + n )
6
n
1
k 3 = ( n 4 + 2n3 + n 2 )
∑
4
k =1
n
1
k 4 = ( 6n5 + 15n4 + 10n3 − n )
∑
30
k =1
∑k
log a N = x ⇒ a x = N
1
3
∞
3
sec
1
y = ∠ sec x = ∠ cos
y ∈ [ 0, π ]
x
1
 π π
y = ∠ csc x = ∠ sen
y ∈ − , 
x
 2 2
Gráfica 1. Las funciones trigonométricas: sen x ,
cos x , tg x :
=
p
2
12
y = ∠ ctg x = ∠ tg
k =1
ap
a
  = p
b
b
2 1
y = ∠ sen x
∑  a + ( k − 1) d  = 2  2a + ( n − 1) d 
2
0
1
3 2 1 3
1
n
ctg
0
12
90
+ bk ) = ∑ ak + ∑ bk
k
= ap ⋅ bp
log a N =
30
tg
cos
k =1
n
2
sen
k =1
n
∑(a
3
1
θ
0
3 2
= c ∑ ak
k
k =1
4
CA
45
n
∑ ca
∑(a
= a pq
p
n
CO
θ
60
k =1
n
a
= a p −q
aq
p q
HIP
n
∑ c = nc
k =1
p+q
p
(a )
par
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x :
n
n
∏a
a+b ≤ a + b ó
impar
+ a n = ∑ ak
a1 + a2 +
a ≥0y a =0 ⇔ a=0
k =1
 n

k +1
( a + b ) ⋅  ∑ ( −1) a n− k b k −1  = a n + b n ∀ n ∈
 k =1

 n

k +1
( a + b ) ⋅  ∑ ( −1) a n− k b k −1  = a n − b n ∀ n ∈
 k =1

SUMAS Y PRODUCTOS
a≤ a y −a≤ a
ab = a b ó
( a + b ) ⋅ ( a 2 − ab + b 2 ) = a3 + b3
( a + b ) ⋅ ( a3 − a 2 b + ab 2 − b3 ) = a 4 − b 4
( a + b ) ⋅ ( a 4 − a 3b + a 2 b 2 − ab3 + b 4 ) = a5 + b5
( a + b ) ⋅ ( a5 − a 4 b + a3b 2 − a 2 b3 + ab4 − b5 ) = a 6 − b6
4
3
2
1
0
-1
-2
senh x
cosh x
tgh x
-3
-4
-5
0
FUNCS HIPERBÓLICAS INVERSAS
(
(
)
)
senh −1 x = ln x + x 2 + 1 , ∀x ∈
cosh −1 x = ln x ± x 2 − 1 , x ≥ 1
tgh −1 x =
1 1+ x 
ln 
,
2 1− x 
ctgh −1 x =
1  x +1
ln 
,
2  x −1
x <1
x >1
 1 ± 1 − x2 
, 0 < x ≤ 1
sech −1 x = ln 


x


2

1
x +1 
−1
, x ≠ 0
csch x = ln  +
x
x 

5
IDENTIDADES DE FUNCS HIP
cosh 2 x − senh 2 x = 1
1 − tgh 2 x = sech 2 x
ctgh 2 x − 1 = csch x
senh ( − x ) = − senh x
cosh ( − x ) = cosh x
tgh ( − x ) = − tgh x
senh ( x ± y ) = senh x cosh y ± cosh x senh y
cosh ( x ± y ) = cosh x cosh y ± senh x senh y
tgh x ± tgh y
1 ± tgh x tgh y
senh 2 x = 2senh x cosh x
tgh ( x ± y ) =
cosh 2 x = cosh 2 x + senh 2 x
2 tgh x
tgh 2 x =
1 + tgh 2 x
1
( cosh 2 x − 1)
2
1
cosh 2 x = ( cosh 2 x + 1)
2
cosh 2 x − 1
tgh 2 x =
cosh 2 x + 1
senh 2 x =
tgh x =
senh 2 x
cosh 2 x + 1
e x = cosh x + senh x
e
−x
= cosh x − senh x
OTRAS
ax 2 + bx + c = 0
−b ± b 2 − 4ac
2a
b 2 − 4ac = discriminante
⇒ x=
exp (α ± i β ) = eα ( cos β ± i sen β ) si α , β ∈
LÍMITES
1
lim (1 + x ) x = e = 2.71828...
x→0
x
 1
lim 1 +  = e
x →∞
x

sen x
=1
lim
x→0
x
1 − cos x
=0
lim
x→0
x
ex − 1
=1
lim
x→0
x
x −1
=1
lim
x →1 ln x
DERIVADAS
Dx f ( x ) =
f ( x + ∆x ) − f ( x )
df
∆y
= lim
= lim
∆x → 0 ∆x
dx ∆x →0
∆x
d
(c) = 0
dx
d
( cx ) = c
dx
d
( cx n ) = ncx n−1
dx
d
du dv dw
(u ± v ± w ± ) = ± ± ±
dx
dx dx dx
d
du
( cu ) = c
dx
dx
d
dv
du
( uv ) = u + v
dx
dx
dx
d
dw
dv
du
( uvw) = uv + uw + vw
dx
dx
dx
dx
d  u  v ( du dx ) − u ( dv dx )
=
 
dx  v 
v2
d n
n −1 du
( u ) = nu dx
dx
dF dF du
(Regla de la Cadena)
=
⋅
dx du dx
du
1
=
dx dx du
dF dF du
=
dx dx du
f 2′ ( t )
dy dy dt
=
=
dx dx dt
f1′( t )
 x = f1 ( t )
donde 
 y = f 2 ( t )
DERIVADA DE FUNCS LOG & EXP
du dx 1 du
d
= ⋅
( ln u ) =
dx
u
u dx
d
log e du
⋅
( log u ) =
dx
u dx
log e du
d
( log a u ) = a ⋅ a > 0, a ≠ 1
dx
u
dx
d u
u du
e
=
e
⋅
( )
dx
dx
d u
du
a ) = a u ln a ⋅
(
dx
dx
d v
du
dv
+ ln u ⋅ u v ⋅
u ) = vu v −1
(
dx
dx
dx
DERIVADA DE FUNCIONES TRIGO
d
du
( sen u ) = cos u
dx
dx
d
du
( cos u ) = − sen u
dx
dx
d
du
( tg u ) = sec2 u
dx
dx
d
du
( ctg u ) = − csc2 u
dx
dx
d
du
( sec u ) = sec u tg u
dx
dx
d
du
( csc u ) = − csc u ctg u
dx
dx
d
du
( vers u ) = sen u
dx
dx
DERIV DE FUNCS TRIGO INVER
1
d
du
⋅
( ∠ sen u ) =
dx
1 − u 2 dx
1
d
du
⋅
( ∠ cos u ) = −
dx
1 − u 2 dx
1
d
du
⋅
( ∠ tg u ) =
dx
1 + u 2 dx
1
d
du
⋅
( ∠ ctg u ) = −
dx
1 + u 2 dx
1
d
du + si u > 1
⋅ 
( ∠ sec u ) = ±
dx
u u 2 − 1 dx − si u < −1
1
d
du  − si u > 1
⋅ 
( ∠ csc u ) = ∓
dx
u u 2 − 1 dx  + si u < −1
du
1
d
⋅
( ∠ vers u ) =
dx
2u − u 2 dx
DERIVADA DE FUNCS HIPERBÓLICAS
d
du
senh u = cosh u
dx
dx
d
du
cosh u = senh u
dx
dx
d
du
tgh u = sech 2 u
dx
dx
d
du
ctgh u = − csch 2 u
dx
dx
d
du
sech u = − sech u tgh u
dx
dx
d
du
csch u = − csch u ctgh u
dx
dx
DERIVADA DE FUNCS HIP INV
d
1
du
senh −1 u =
⋅
dx
1 + u 2 dx
-1
+
d
±1
du
 si cosh u > 0
cosh −1 u =
⋅ , u >1 
-1
dx
u 2 − 1 dx
− si cosh u < 0
d
1
du
⋅ , u <1
tgh −1 u =
dx
1 − u 2 dx
d
1
du
⋅ , u >1
ctgh −1 u =
dx
1 − u 2 dx
−1
∓1
d
du − si sech u > 0, u ∈ 0,1
⋅ 
sech −1 u =
−1
dx
u 1 − u 2 dx + si sech u < 0, u ∈ 0,1
d
1
du
csch −1 u = −
⋅ , u≠0
dx
u 1 + u 2 dx
INTEGRALES DEFINIDAS,
PROPIEDADES
∫
∫
∫
∫
∫
b
a
b
{ f ( x ) ± g ( x )} dx = ∫ f ( x ) dx ± ∫ g ( x ) dx
b
b
a
a
b
a
cf ( x ) dx = c ⋅ ∫ f ( x ) dx
b
c
a
b
a
a
a
c∈
a
b
f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx
a
c
a
f ( x ) dx = − ∫ f ( x ) dx
b
f ( x ) dx = 0
b
m ⋅ ( b − a ) ≤ ∫ f ( x ) dx ≤ M ⋅ ( b − a )
a
⇔ m ≤ f ( x ) ≤ M ∀x ∈ [ a, b ] , m, M ∈
INTEGRALES DE FUNCS LOG & EXP
∫ e du = e
u
u
a u a > 0
∫ a du = ln a a ≠ 1
u
au 
−1
1 
∫ ua du = ln a ⋅  u − ln a 
u
1
= ln tgh u
2
∫ ue du = e ( u − 1)
∫ ln udu =u ln u − u = u ( ln u − 1)
u
u
INTREGRALES DE FRAC
1
u
∫ log a udu =ln a ( u ln u − u ) = ln a ( ln u − 1)
u2
∫ u log a udu = 4 ⋅ ( 2 log a u − 1)
u2
∫ u ln udu = 4 ( 2 ln u − 1)
INTEGRALES DE FUNCS TRIGO
∫ sen udu = − cos u
∫ cos udu = sen u
∫ sec udu = tg u
∫ csc udu = − ctg u
∫ sec u tg udu = sec u
∫ csc u ctg udu = − csc u
∫ tg udu = − ln cos u = ln sec u
∫ ctg udu = ln sen u
∫ sec udu = ln sec u + tg u
∫ csc udu = ln csc u − ctg u
∫ ctg
2
2
b
INTEGRALES
∫ adx =ax
∫ af ( x ) dx = a ∫ f ( x ) dx
∫ ( u ± v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ±
∫ udv = uv − ∫ vdu ( Integración por partes )
u n +1
∫ u du = n + 1 n ≠ −1
du
∫ u = ln u
n
udu =
∫e
∫ u sen udu = sen u − u cos u
du
a2 ± u 2
du
< a2 )
u
a
)
au
sen bu du =
)
e au ( a sen bu − b cos bu )
a2 + b2
e au ( a cos bu + b sen bu )
a2 + b2
ALGUNAS SERIES
2
2
+
+
f ( n ) ( x0 )( x − x0 )
n!
f ( x ) = f ( 0) + f '( 0) x +
2
2
−1
= u∠ sec u − ∠ cosh u
∫ ∠ csc udu = u∠ csc u + ln ( u +
u2 − 1
= u∠ csc u + ∠ cosh u
INTEGRALES DE FUNCS HIP
)
)
f '' ( x0 )( x − x0 )
f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) +
2
2
2
=
au
∫ e cos bu du =
∫ ∠ sen udu = u∠ sen u + 1 − u
∫ ∠ cos udu = u∠ cos u − 1 − u
∫ ∠ tg udu = u∠ tg u − ln 1 + u
∫ ∠ ctg udu = u∠ ctg u + ln 1 + u
∫ ∠ sec udu = u∠ sec u − ln ( u + u
2
(u
(
udu = − ( ctg u + u )
∫ senh udu = cosh u
∫ cosh udu = senh u
∫ sech udu = tgh u
∫ csch udu = − ctgh u
∫ sech u tgh udu = − sech u
∫ csch u ctgh udu = − csch u
> a2 )
1
u
ln
a a + a2 ± u 2
1
a
∫ u u 2 − a 2 = a ∠ cos u
1
u
= ∠ sec
a
a
u 2
a2
u
2
2
2
∫ a − u du = 2 a − u + 2 ∠ sen a
2
u 2
a
2
2
2
2
2
∫ u ± a du = 2 u ± a ± 2 ln u + u ± a
MAS INTEGRALES
INTEGRALES DE FUNCS TRIGO INV
a
2
= ln u + u 2 ± a 2
u 2 ± a2
∫u
a
b
(
du
∫
u 1
− sen 2u
2 4
u 1
2
∫ cos udu = 2 + 4 sen 2u
2
∫ tg udu = tg u − u
∫ sen
(u
u
a
= −∠ cos
a
a
= ∠ sen
a2 − u2
2
∫ u cos udu = cos u + u sen u
∫ f ( x ) dx ≤ ∫ f ( x ) dx si a < b
2
du
∫
2
b
⇔ f ( x ) ≤ g ( x ) ∀x ∈ [ a , b ]
1
du
u
= ∠ tg
a
+ a2 a
1
u
= − ∠ ctg
a
a
du
1 u−a
∫ u 2 − a 2 = 2a ln u + a
du
1 a+u
∫ a 2 − u 2 = 2a ln a − u
∫u
INTEGRALES CON
b
∫ f ( x ) dx ≤ ∫ g ( x ) dx
∫ tgh udu = ln cosh u
∫ ctgh udu = ln senh u
∫ sech udu = ∠ tg ( senh u )
∫ csch udu = − ctgh ( cosh u )
+
+
f
( n)
( 0) x
2!
n
f '' ( 0 ) x
: Taylor
2
2!
n
: Maclaurin
n!
x 2 x3
xn
+
+ +
+
2! 3!
n!
3
5
7
x
x
x
x 2 n −1
n −1
sen x = x − +
−
+ + ( −1)
3! 5! 7!
( 2n − 1)!
ex = 1 + x +
cos x = 1 −
x2 x4 x6
+
−
+
2! 4! 6!
+ ( −1)
n −1
x 2n−2
( 2n − 2 ) !
n
x 2 x3 x 4
n −1 x
+
−
+ + ( −1)
2
3
4
n
2 n −1
x3 x5 x7
n −1 x
∠ tg x = x − +
−
+ + ( −1)
3
5
7
2n − 1
ln (1 + x ) = x −
2