This document provides formulas for calculus and trigonometry. It includes formulas for: 1) Factoring polynomials with the difference of squares formula like (a + b)(a^2 - ab + b^2) = a^3 + b^3. 2) Trigonometric identities for sine, cosine, tangent, cotangent, secant, and cosecant functions. 3) Formulas for derivatives of polynomials using the binomial theorem.

1. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
Formulario de ( a + b ) ⋅ ( a 2 − ab + b2 ) = a3 + b3 θ sen cos tg ctg sec csc
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x : sen α + sen β
1 1
= 2sen (α + β ) ⋅ cos (α − β )
( a + b ) ⋅ ( a3 − a 2 b + ab2 − b3 ) = a 4 − b4 0 0 ∞ ∞ 0 2 2
Cálculo Diferencial
1 1
1 1
30 12 3 2 3 2 sen α − sen β = 2 sen (α − β ) ⋅ cos (α + β )
4
3 2 1 3
( a + b ) ⋅ ( a 4 − a3b + a 2 b2 − ab3 + b4 ) = a5 + b5
e Integral VER.4.3 45 1 2 1 2 1 1 2 2 3
2 2
( a + b ) ⋅ ( a5 − a 4 b + a3b2 − a 2 b3 + ab4 − b5 ) = a 6 − b6
1 1
60 3 2 12 3 1 3 2 2 3 cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β )
Jesús Rubí Miranda (jesusrubim@yahoo.com) 2 2 2
90 1 0 ∞ 0 ∞ 1
http://mx.geocities.com/estadisticapapers/ ⎛ n ⎞ cos α − cos β
1 1
= −2sen (α + β ) ⋅ sen (α − β )
( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈
k +1
impar ⎡ π π⎤ 1
http://mx.geocities.com/dicalculus/ ⎝ k =1 ⎠ y = ∠ sen x y ∈ ⎢− , ⎥ 2 2
⎣ 2 2⎦
sen (α ± β )
0
VALOR ABSOLUTO ⎛ n
⎞
( a + b ) ⋅ ⎜ ∑ ( −1) y ∈ [ 0, π ]
k +1
a n − k b k −1 ⎟ = a n − b n ∀ n ∈ par y = ∠ cos x tg α ± tg β =
⎧a si a ≥ 0 ⎝ k =1 ⎠
-1
cos α ⋅ cos β
a =⎨
arc ctg x
π π arc sec x
⎩− a si a < 0 y = ∠ tg x y∈ − arc csc x
SUMAS Y PRODUCTOS , 1
⎡sen (α − β ) + sen (α + β ) ⎤
-2
2 2 -5 0 5
sen α ⋅ cos β =
a = −a n
2⎣ ⎦
a1 + a2 + + an = ∑ ak 1 IDENTIDADES TRIGONOMÉTRICAS
a ≤ a y −a≤ a y = ∠ ctg x = ∠ tg y ∈ 0, π 1
k =1
x sen α ⋅ sen β = ⎡cos (α − β ) − cos (α + β ) ⎤
sen θ + cos2 θ = 1 2⎣ ⎦
2
n
a ≥0y a =0 ⇔ a=0
∑ c = nc y = ∠ sec x = ∠ cos
1
y ∈ [ 0, π ] 1 + ctg 2 θ = csc2 θ 1
k =1 cos α ⋅ cos β = ⎡cos (α − β ) + cos (α + β ) ⎤
2⎣ ⎦
n n x
ab = a b ó ∏a = ∏ ak n n
⎡ π π⎤ tg 2 θ + 1 = sec2 θ
∑ ca = c ∑ ak 1
k
k =1 k =1 k y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥ tg α + tg β
n n k =1 k =1 x ⎣ 2 2⎦ sen ( −θ ) = − sen θ tg α ⋅ tg β =
a+b ≤ a + b ó ∑a ≤ ∑ ak n n n ctg α + ctg β
k =1
k
k =1 ∑(a
k =1
k + bk ) = ∑ ak + ∑ bk
k =1 k =1
Gráfica 1. Las funciones trigonométricas: sen x , cos ( −θ ) = cosθ FUNCIONES HIPERBÓLICAS
cos x , tg x :
EXPONENTES n tg ( −θ ) = − tg θ e x − e− x
a p ⋅ a q = a p+q ∑(a − ak −1 ) = an − a0 senh x =
sen (θ + 2π ) = sen θ
k 2
k =1
2
ap 1.5
e x + e− x
= a p−q n
n cos (θ + 2π ) = cosθ cosh x =
aq ∑ ⎡ a + ( k − 1) d ⎤ = 2 ⎡ 2a + ( n − 1) d ⎤
⎣ ⎦ ⎣ ⎦
1
2
k =1 tg (θ + 2π ) = tg θ
( a p ) = a pq senh x e x − e − x
q 0.5
n tgh x = =
(a + l )
= 0 sen (θ + π ) = − sen θ cosh x e x + e − x
(a ⋅ b) = ap ⋅bp
p
2 -0.5
cos (θ + π ) = − cosθ 1 e x + e− x
n
1 − r n a − rl ctgh x = =
⎛a⎞ ap
p
∑ ar k −1
=a = -1
tg (θ + π ) = tg θ tgh x e x − e − x
⎜ ⎟ = p k =1 1− r 1− r
⎝b⎠
sen x
b -1.5
1 2
sen (θ + nπ ) = ( −1) sen θ sech x = =
cos x n
( n + n)
n
1 2
∑k =
tg x
a p/q
= a
q p -2
-8 -6 -4 -2 0 2 4 6 8 cosh x e x + e − x
2
k =1
cos (θ + nπ ) = ( −1) cos θ
n
1 2
LOGARITMOS csch x = =
∑ k 2 = 6 ( 2n3 + 3n2 + n )
n
1 Gráfica 2. Las funciones trigonométricas csc x ,
log a N = x ⇒ a x = N tg (θ + nπ ) = tg θ senh x e x − e − x
k =1 sec x , ctg x :
log a MN = log a M + log a N senh : →
sen ( nπ ) = 0
∑ k 3 = 4 ( n 4 + 2n3 + n 2 )
n
1
M
2.5
cosh : → [1, ∞
= log a M − log a N cos ( nπ ) = ( −1)
n
log a k =1 2
N tgh : → −1,1
∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n )
n
1
tg ( nπ ) = 0
1.5
log a N r = r log a N 1
ctgh : − {0} → −∞ , −1 ∪ 1, ∞
k =1
⎛ 2n + 1 ⎞
+ ( 2n − 1) = n 2 π ⎟ = ( −1) → 0,1]
0.5 n
log b N ln N 1+ 3 + 5 + sen ⎜
log a N = = sech :
⎝ 2 ⎠
0
log b a ln a n
− {0} → − {0}
n! = ∏ k
-0.5
csch :
log10 N = log N y log e N = ln N ⎛ 2n + 1 ⎞
k =1
-1
cos ⎜ π⎟=0
ALGUNOS PRODUCTOS -1.5 ⎝ 2 ⎠ Gráfica 5. Las funciones hiperbólicas senh x ,
⎛n⎞ n! csc x
a ⋅ ( c + d ) = ac + ad ⎜ ⎟= , k≤n -2 sec x
⎛ 2n + 1 ⎞ cosh x , tgh x :
⎝ k ⎠ ( n − k )!k !
ctg x
tg ⎜ π⎟=∞
⎝ 2 ⎠
-2.5
( a + b) ⋅ ( a − b) = a − b
-8 -6 -4 -2 0 2 4 6 8
2 2
5
n
⎛n⎞
( x + y ) = ∑ ⎜ ⎟ xn−k y k π⎞
n
Gráfica 3. Las funciones trigonométricas inversas ⎛
( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b2
4
sen θ = cos ⎜θ − ⎟
2
k =0 ⎝ k ⎠ arcsen x , arccos x , arctg x : ⎝ 2⎠ 3
( a − b ) ⋅ ( a − b ) = ( a − b ) = a 2 − 2ab + b 2
2 2
( x1 + x2 + + xk )
n
=∑
n!
x1n1 ⋅ x2 2 ⎛ π⎞
cosθ = sen ⎜θ + ⎟
n nk
x 4
( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd
1
k
n1 !n2 ! nk !
3
⎝ 2⎠ 0
( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd CONSTANTES sen (α ± β ) = sen α cos β ± cos α sen β -1
( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd π = 3.14159265359… 2
cos (α ± β ) = cos α cos β ∓ sen α sen β
-2
se nh x
e = 2.71828182846…
-3 co sh x
( a + b ) = a3 + 3a 2b + 3ab2 + b3 tg α ± tg β
3 1 tgh x
tg (α ± β ) =
-4
-5 0 5
TRIGONOMETRÍA
1 ∓ tg α tg β
( a − b ) = a3 − 3a 2b + 3ab2 − b3
3 0
CO 1 FUNCIONES HIPERBÓLICAS INV
sen θ = cscθ = sen 2θ = 2sen θ cosθ
( a + b + c ) = a 2 + b2 + c 2 + 2ab + 2ac + 2bc
2 HIP
CA
sen θ
1
-1
arc sen x
arc cos x
arc tg x cos 2θ = cos 2 θ − sen 2 θ (
senh −1 x = ln x + x 2 + 1 , ∀x ∈ )
cosθ = secθ =
( a − b ) ⋅ ( a + ab + b ) = a − b ( )
-2
2 tg θ
2 2 3 3
cosθ
-3 -2 -1 0 1 2 3
HIP tg 2θ = cosh −1 x = ln x ± x 2 − 1 , x ≥ 1
sen θ CO 1 − tg 2 θ
( a − b ) ⋅ ( a3 + a 2 b + ab2 + b3 ) = a 4 − b4 tg θ = = ctg θ =
1
1 ⎛1+ x ⎞
cosθ CA tg θ 1 tgh −1 x = ln ⎜ ⎟, x < 1
( a − b ) ⋅ ( a 4 + a3b + a 2 b2 + ab3 + b 4 ) = a5 − b5 sen 2 θ = (1 − cos 2θ ) 2 ⎝ 1− x ⎠
2
⎛ ⎞ π radianes=180 1 ⎛ x +1 ⎞
ctgh −1 x = ln ⎜
n
1
( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n ∀n ∈ cos 2 θ = (1 + cos 2θ ) ⎟, x > 1
2 ⎝ x −1 ⎠
⎝ k =1 ⎠ 2
1 − cos 2θ ⎛ 1 ± 1 − x2 ⎞
HIP tg 2 θ = sech −1 x = ln ⎜ ⎟, 0 < x ≤ 1
CO 1 + cos 2θ ⎜ x ⎟
⎝ ⎠
θ ⎛1 x2 + 1 ⎞
csch −1 x = ln ⎜ + ⎟, x ≠ 0
CA ⎜x x ⎟
⎝ ⎠

2. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
∫ tgh udu = ln cosh u
IDENTIDADES DE FUNCS HIP d dv du DERIVADA DE FUNCS HIPERBÓLICAS INTEGRALES DE FUNCS LOG & EXP
( uv ) = u + v
cosh 2 x − senh 2 x = 1 d du
∫ e du = e
u u
dx dx dx senh u = cosh u
1 − tgh 2 x = sech 2 x d dw dv du dx dx ∫ ctgh udu = ln senh u
( uvw) = uv + uw + vw a u ⎧a > 0
∫ a du = ln a ⎨a ≠ 1 ∫ sech udu = ∠ tg ( senh u )
d du u
dx dx dx dx cosh u = senh u
ctgh x − 1 = csch x
2
⎩
d ⎛ u ⎞ v ( du dx ) − u ( dv dx ) dx dx
senh ( − x ) = − senh x ∫ csch udu = − ctgh ( cosh u )
−1
⎜ ⎟= d du au ⎛ 1 ⎞
dx ⎝ v ⎠ v2 tgh u = sech 2 u ∫ ua du = ln a ⋅ ⎜ u − ln a ⎟
u
cosh ( − x ) = cosh x dx dx ⎝ ⎠ 1
= ln tgh u
d n
( u ) = nu n−1 du
tgh ( − x ) = − tgh x d du
∫ ue du = e ( u − 1) 2
u u
dx dx ctgh u = − csch 2 u
dx dx INTEGRALES DE FRAC
senh ( x ± y ) = senh x cosh y ± cosh x senh y dF dF du ∫ ln udu =u ln u − u = u ( ln u − 1)
= ⋅ (Regla de la Cadena) d
sech u = − sech u tgh u
du du 1 u
cosh ( x ± y ) = cosh x cosh y ± senh x senh y dx du dx dx dx 1 u ∫ u 2 + a 2 = a ∠ tg a
du 1 ∫ log ( u ln u − u ) = ( ln u − 1)
udu =
tgh x ± tgh y =
a
d du ln a ln a 1 u
tgh ( x ± y ) = dx dx du csch u = − csch u ctgh u = − ∠ ctg
1 ± tgh x tgh y dx dx u 2
∫ u log a udu = 4 ⋅ ( 2log a u − 1)
a a
dF dF du DERIVADA DE FUNCS HIP INV 1 u−a
senh 2 x = 2senh x cosh x
∫ u 2 − a 2 2a u + a ( u > a )
= du
= ln 2 2
dx dx du d 1 du u2
cosh 2 x = cosh x + senh x senh −1 u = ⋅
∫ u ln udu = 4 ( 2ln u − 1)
2 2
dy dy dt f 2′ ( t ) ⎪ x = f1 ( t )
⎧ 1 + u 2 dx 1 a+u
∫ a 2 − u 2 = 2a ln a − u ( u < a )
dx du 2 2
tgh 2 x =
2 tgh x = = donde ⎨
1 + tgh 2 x dx dx dt f1′( t ) ⎪ y = f2 ( t )
⎩ d ±1 du ⎧+ si cosh -1u > 0
⎪ INTEGRALES DE FUNCS TRIGO
cosh −1 u = ⋅ , u >1 ⎨
1 DERIVADA DE FUNCS LOG & EXP dx u 2 − 1 dx ⎪− si cosh u < 0
⎩
-1
∫ sen udu = − cos u INTEGRALES CON
senh 2 x = ( cosh 2 x − 1)
2 d
( ln u ) =
du dx 1 du
= ⋅
d
tgh −1 u =
1 du
⋅ , u <1 ∫ cos udu = sen u ∫
du
= ∠ sen
u
1 dx u u dx dx 1 − u 2 dx a2 − u2 a
cosh x = ( cosh 2 x + 1)
2
d log e du d 1 du ∫ sec udu = tg u
2
2 ( log u ) = ⋅ ctgh −1 u = ⋅ , u >1 = −∠ cos
u
cosh 2 x − 1 1 − u 2 dx ∫ csc udu = − ctg u
2
dx u dx dx a
tgh 2 x =
⎧−
( )
−1
cosh 2 x + 1 d log e du
( log a u ) = a ⋅ a > 0, a ≠ 1 ∓1 du ⎪ si sech u > 0, u ∈ 0,1 du
∫ sec u tg udu = sec u
d
senh 2 x dx u dx dx
sech −1 u = ⋅ ⎨ −1
u 1 − u 2 dx ⎪ + si sech u < 0, u ∈ 0,1
∫ u 2 ± a2
= ln u + u 2 ± a 2
tgh x = ⎩
cosh 2 x + 1
d u
( e ) = eu ⋅ du d 1 du ∫ csc u ctg udu = − csc u du 1 u
dx dx −1
csch u = − ⋅ , u≠0 ∫u = ln
e x = cosh x + senh x dx u 1 + u 2 dx ∫ tg udu = − ln cos u = ln sec u a2 ± u2
a a + a2 ± u 2
e − x = cosh x − senh x
d u
( a ) = au ln a ⋅ du du 1 a
dx dx INTEGRALES DEFINIDAS, PROPIEDADES ∫ ctg udu = ln sen u ∫ u u 2 − a 2 = a ∠ cos u
OTRAS d v
( u ) = vu v−1 du + ln u ⋅ u v ⋅ dv ∫ { f ( x ) ± g ( x )} dx = ∫ f ( x ) dx ± ∫ g ( x ) dx
b b b
ax + bx + c = 0
2
dx dx dx a a a ∫ sec udu = ln sec u + tg u 1
= ∠ sec
u
∫ cf ( x ) dx = c ⋅ ∫ f ( x ) dx c ∈ ∫ csc udu = ln csc u − ctg u
b b
DERIVADA DE FUNCIONES TRIGO a a
−b ± b 2 − 4ac
⇒ x= d du
a a
u 2 a2 u
( sen u ) = cos u ∫ a − u du = 2 a − u + 2 ∠ sen a
∫ f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx
b c b 2 2 2
2a u 1
∫ sen udu =
− sen 2u
2
dx dx
b 2 − 4ac = discriminante a a c
2 4
( )
2
d du
( cos u ) = − sen u ∫ f ( x ) dx = −∫ f ( x ) dx
b a
u 2 a
exp (α ± iβ ) = eα ( cos β ± i sen β ) si α , β ∈ ∫ u ± a du = 2 u ± a ± 2 ln u + u ± a
2 2 2 2 2
u 1
∫ cos udu = 2 + 4 sen 2u
2
dx dx a b
∫ f ( x ) dx = 0
a
LÍMITES d du
( tg u ) = sec2 u MÁS INTEGRALES
∫ tg udu = tg u − u
a 2
e au ( a sen bu − b cos bu )
1
lim (1 + x ) x = e = 2.71828... dx dx
m ⋅ ( b − a ) ≤ ∫ f ( x ) dx ≤ M ⋅ ( b − a )
b
∫e sen bu du =
au
∫ ctg udu = − ( ctg u + u )
x →0 d du
( ctg u ) = − csc2 u a 2 + b2
2
a
x
⎛ 1⎞
lim ⎜1 + ⎟ = e
dx dx ⇔ m ≤ f ( x ) ≤ M ∀x ∈ [ a, b ] , m, M ∈ e au ( a cos bu + b sen bu )
⎝ x⎠ ∫ u sen udu = sen u − u cos u ∫ e cos bu du =
x →∞ au
d du
( sec u ) = sec u tg u ∫ f ( x ) dx ≤ ∫ g ( x ) dx
b b
a 2 + b2
sen x dx dx
=1 ∫ u cos udu = cos u + u sen u
a a
lim ALGUNAS SERIES
x →0 d du ⇔ f ( x ) ≤ g ( x ) ∀x ∈ [ a, b ]
x
( csc u ) = − csc u ctg u INTEGRALES DE FUNCS TRIGO INV f '' ( x0 )( x − x0 )
2
1 − cos x dx dx f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) +
=0
∫ f ( x ) dx ≤ ∫ f ( x ) dx si a < b
b b
lim
∫ ∠ sen udu = u∠ sen u + 1 − u 2!
2
x →0 d du
x
( vers u ) = sen u a a
f(
n)
( x0 )( x − x0 )
n
ex −1 dx dx INTEGRALES
=1 ∫ ∠ cos udu = u∠ cos u − 1 − u + +
2
lim : Taylor
∫ adx =ax
x →0 x DERIV DE FUNCS TRIGO INVER n!
x −1 ∫ ∠ tg udu = u∠ tg u − ln 1 + u f '' ( 0 ) x
2 2
d 1 du
lim
x →1 ln x
=1 ( ∠ sen u ) = ⋅
∫ af ( x ) dx = a ∫ f ( x ) dx f ( x ) = f ( 0) + f '( 0) x +
dx 1 − u 2 dx 2!
∫ ∠ ctg udu = u∠ ctg u + ln 1 + u
2
∫ ( u ± v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ± ( 0) x
DERIVADAS ( n) n
d 1 du f
∫ ∠ sec udu = u∠ sec u − ln ( u + u )
df f ( x + ∆x ) − f ( x ) ∆y
( ∠ cos u ) = − ⋅ 2
−1 + + : Maclaurin
Dx f ( x ) = dx 1 − u 2 dx
∫ udv = uv − ∫ vdu ( Integración por partes )
= lim = lim n!
dx ∆x→0 ∆x ∆x → 0 ∆x
d 1 du = u∠ sec u − ∠ cosh u x 2 x3 xn
( ∠ tg u ) = ⋅ u n +1 ex = 1 + x + + + + +
∫ ∠ csc udu = u∠ csc u + ln ( u + )
d
(c) = 0 dx 1 + u 2 dx ∫ u du = n + 1
n
n ≠ −1
u 2 −1
2! 3! n!
dx d 1 du x3 x 5 x 7 x 2 n −1
( ∠ ctg u ) = − 2 ⋅ sen x = x − + − + + ( −1)
n −1
d
( cx ) = c dx 1 + u dx
du
∫ u = ln u = u∠ csc u + ∠ cosh u 3! 5! 7! ( 2n − 1)!
dx
d 1 du ⎧ + si u > 1 INTEGRALES DE FUNCS HIP
x 2n−2
( ∠ sec u ) = ± 2 ⋅ ⎨ cos x = 1 −
x2 x4 x6
+ − + + ( −1)
n −1
d
dx
( cx n ) = ncxn−1 dx u u − 1 dx ⎩ − si u < −1 ∫ senh udu = cosh u 2! 4! 6! ( 2n − 2 )!
d 1 du ⎧− si u > 1 ∫ cosh udu = senh u
( ∠ csc u ) = ∓ x 2 x3 x 4 n
d du dv dw n −1 x
(u ± v ± w ± ) = ± ± ± ⋅ ⎨ ln (1 + x ) = x −+ − + + ( −1)
dx u u 2 − 1 dx ⎩+ si u < −1 2 3 4 n
∫ sech udu = tgh u
dx dx dx dx 2
2 n −1
d du d 1 du x3 x5 x7
( cu ) = c ( ∠ vers u ) = ⋅ ∠ tg x = x − + − + + ( −1)
n −1 x
∫ csch udu = − ctgh u
2
dx dx dx 2u − u 2 dx 3 5 7 2n − 1
∫ sech u tgh udu = − sech u
∫ csch u ctgh udu = − csch u

3. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
ÁLGEBRA LINEAL
Def. El determinante de una matriz
⎡a a ⎤
A = ⎢ 11 12 ⎥
⎣ a21 a22 ⎦
está dado por
a11 a12
det A = = a11a22 − a12 a21 .
a21 a22
Def. El determinante de una matriz
⎡ a11 a12 a13 ⎤
⎢ ⎥
A = ⎢ a21 a22 a23 ⎥
⎢
⎣ a31 a32 ⎥
a33 ⎦
está dado por
a11 a12 a13 a11 ⋅ a22 ⋅ a33 + a12 ⋅ a23 ⋅ a31
det A = a21 a22 a23 = + a13 ⋅ a21 ⋅ a32 − a11 ⋅ a23 ⋅ a32 .
a31 a32 a33 −a12 ⋅ a21 ⋅ a33 − a13 ⋅ a22 ⋅ a31