1. Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
Fórmulas de ( a + b ) ⋅ ( a 2 − ab + b 2 ) = a 3 + b3 θ sin cos tg ctg sec csc Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x : sin α + sin β = 2sin
1 1
(α + β ) ⋅ cos (α − β )
∞ ∞
( a + b ) ⋅ ( a3 − a 2 b + ab 2 − b3 ) = a 4 − b 4
0 0 1 0 1 2 2
Cálculo Diferencial 30 12 3 2 1 3 3 2 3 2 4 1 1
sin α − sin β = 2 sin (α − β ) ⋅ cos (α + β )
( a + b ) ⋅ ( a 4 − a 3b + a 2 b 2 − ab3 + b 4 ) = a 5 + b5
e Integral VER.6.8 45 1 2 1 2 1 1 2 2 3
2 2
( a + b ) ⋅ ( a5 − a 4 b + a 3b 2 − a 2 b3 + ab 4 − b5 ) = a 6 − b 6
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) 90 1 0 ∞ 0 ∞ 1
2 2 2
http://www.geocities.com/calculusjrm/ ⎛ n ⎞ 1 1
cos α − cos β = −2 sin (α + β ) ⋅ sin (α − β )
( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈
k +1
⎡ π π⎤ 1
impar y = ∠ sin x y ∈ ⎢− , ⎥ 2 2
⎝ k =1 ⎠ ⎣ 2 2⎦
VALOR ABSOLUTO
sin (α ± β )
0
⎛ ⎞ y = ∠ cos x y ∈ [ 0, π ]
n
⎧a si a ≥ 0 ( a + b ) ⋅ ⎜ ∑ ( −1)
k +1
a n − k b k −1 ⎟ = a n − b n ∀ n ∈ par tg α ± tg β =
a =⎨ ⎝ k =1 ⎠
-1
arc ctg x cos α ⋅ cos β
⎩− a si a < 0 π π arc sec x
y = ∠ tg x y∈ − , arc csc x
1
SUMAS Y PRODUCTOS sin α ⋅ cos β = ⎡sin (α − β ) + sin (α + β ) ⎤
-2
a = −a 2 2
2⎣ ⎦
-5 0 5
n
a ≤ a y −a ≤ a a1 + a2 + + an = ∑ ak y = ∠ ctg x = ∠ tg
1
y ∈ 0, π IDENTIDADES TRIGONOMÉTRICAS 1
k =1 x sin α ⋅ sin β = ⎡cos (α − β ) − cos (α + β ) ⎤
sin θ + cos 2 θ = 1 2⎣ ⎦
2
a ≥0 y a =0 ⇔ a=0 n
∑ c = nc
1
y = ∠ sec x = ∠ cos y ∈ [ 0, π ] 1 + ctg 2 θ = csc 2 θ 1
n n
x cos α ⋅ cos β = ⎡cos (α − β ) + cos (α + β ) ⎤
∏a = ∏ ak 2⎣ ⎦
k =1
ab = a b ó tg 2 θ + 1 = sec 2 θ
k n n
1 ⎡ π π⎤
k =1 k =1
∑ ca = c ∑ ak y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥ tg α + tg β
⎣ 2 2⎦ sin ( −θ ) = − sin θ
k
n n k =1 k =1 x tg α ⋅ tg β =
a+b ≤ a + b ó ∑a ≤ ∑ ak n n n ctg α + ctg β
cos ( −θ ) = cos θ
k
k =1 k =1
∑ ( ak + bk ) = ∑ ak + ∑ bk Gráfica 1. Las funciones trigonométricas: sin x ,
FUNCIONES HIPERBÓLICAS
cos x , tg x :
tg ( −θ ) = − tg θ
k =1 k =1 k =1
EXPONENTES
ex − e− x
sinh x =
n
a p ⋅ a q = a p+q ∑(a
k =1
k − ak −1 ) = an − a0 2
sin (θ + 2π ) = sin θ 2
ap e x + e− x
= a p−q
1.5
n
n cos (θ + 2π ) = cos θ cosh x =
aq
∑ ⎡ a + ( k − 1) d ⎤ = 2 ⎡ 2a + ( n − 1) d ⎤
⎣ ⎦ ⎣ ⎦
1
tg (θ + 2π ) = tg θ
2
(a )
p q
=a pq k =1 0.5
tgh x =
sinh x e x − e − x
=
n
= (a + l ) sin (θ + π ) = − sin θ cosh x e x + e− x
(a ⋅b)
0
= a ⋅b
p p p
2 -0.5
cos (θ + π ) = − cos θ 1 e x + e− x
p n
1 − r n a − rl ctgh x = =
⎛a⎞ ap
⎜ ⎟ = p ∑ ar = a 1 − r = 1 − r
k −1 -1
tg (θ + π ) = tg θ tgh x e x − e − x
⎝b⎠ b k =1
-1.5 sen x
1 2
sin (θ + nπ ) = ( −1) sin θ sech x = =
cos x n
∑ k = 2 ( n2 + n )
n
1 tg x
a p/q = a p cosh x e x + e − x
q
-2
-8 -6 -4 -2 0 2 4 6 8
cos (θ + nπ ) = ( −1) cos θ
n
k =1
LOGARITMOS 1 2
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 sinh x e x − e − x
sec x , ctg x :
log a MN = log a M + log a N k =1
sinh : →
sin ( nπ ) = 0
∑ k 3 = 4 ( n 4 + 2n3 + n 2 )
n
1
M 2.5
cosh : → [1, ∞
log a = log a M − log a N k =1 2 cos ( nπ ) = ( −1)
n
N tgh : → −1,1
∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n ) tg ( nπ ) = 0
n
1 1.5
log a N r = r log a N 1 ctgh : − {0} → −∞ , −1 ∪ 1, ∞
k =1
⎛ 2n + 1 ⎞
π ⎟ = ( −1)
log b N ln N
→ 0 ,1]
n
+ ( 2n − 1) = n
0.5
log a N = = 1+ 3 + 5 + 2 sin ⎜ sech :
log b a ln a 0
⎝ 2 ⎠
n csch : − {0} → − {0}
n! = ∏ k ⎛ 2n + 1 ⎞
-0.5
log10 N = log N y log e N = ln N -1 cos ⎜ π⎟=0
ALGUNOS PRODUCTOS k =1
-1.5 ⎝ 2 ⎠ Gráfica 5. Las funciones hiperbólicas sinh x ,
a ⋅ ( c + d ) = ac + ad ⎛n⎞ n! csc x
⎛ 2n + 1 ⎞
⎜ ⎟= , k≤n cosh x , tgh x :
-2
π⎟=∞
sec x
⎝ k ⎠ ( n − k )!k !
ctg x tg ⎜
( a + b) ⋅ ( a − b) = a − b 2 2
-2.5
-8 -6 -4 -2 0 2 4 6 8 ⎝ 2 ⎠ 5
n
⎛n⎞ π⎞
4
( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2 ( x + y ) = ∑ ⎜ ⎟ xn−k y k ⎛
n
2 Gráfica 3. Las funciones trigonométricas inversas
sin θ = cos ⎜ θ − ⎟ 3
k =0 ⎝ k ⎠ arcsin x , arccos x , arctg x : ⎝ 2⎠
( a − b ) ⋅ ( a − b ) = ( a − b ) = a 2 − 2ab + b 2
2
2
n! ⎛ π⎞
( x1 + x2 + + xk ) = ∑
1
x1n1 ⋅ x2 2 cos θ = sin ⎜ θ + ⎟
n
( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd
n
xknk 4
⎝ 2⎠ 0
n1 ! n2 ! nk !
( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd
-1
sin (α ± β ) = sin α cos β ± cos α sin β
3
CONSTANTES -2
( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd π = 3.14159265359… 2
cos (α ± β ) = cos α cos β ∓ sin α sin β -3
senh x
cosh x
tgh x
( a + b ) = a3 + 3a 2b + 3ab 2 + b3
3 e = 2.71828182846… 1
tg α ± tg β
-4
-5 0 5
TRIGONOMETRÍA tg (α ± β ) = FUNCIONES HIPERBÓLICAS INV
( a − b ) = a 3 − 3a 2b + 3ab 2 − b3
3 1 ∓ tg α tg β
( )
0
sen θ =
CO
cscθ =
1
sin 2θ = 2sin θ cos θ sinh −1 x = ln x + x 2 + 1 , ∀x ∈
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
2
HIP sen θ
( )
-1
cos 2θ = cos 2 θ − sin 2 θ
arc sen x
cosh −1 x = ln x ± x 2 − 1 , x ≥ 1
arc cos x
( a − b ) ⋅ ( a + ab + b ) = a − b
CA 1 arc tg x
2 2 3 3
cosθ = secθ = -2
2 tg θ
cosθ
-3 -2 -1 0 1 2 3
HIP tg 2θ = 1 ⎛1+ x ⎞
( a − b ) ⋅ ( a 3 + a 2 b + ab 2 + b3 ) = a 4 − b 4 sen θ CO 1 1 − tg 2 θ tgh −1 x = ln ⎜ ⎟, x <1
tgθ = = ctgθ = 2 ⎝1− x ⎠
( a − b ) ⋅ ( a 4 + a 3b + a 2 b 2 + ab3 + b 4 ) = a 5 − b5 cosθ CA tgθ 1
sin 2 θ = (1 − cos 2θ ) 1 ⎛ x +1⎞
2 ctgh −1 x = ln ⎜ ⎟, x >1
⎛ n
⎞ π radianes=180 2 ⎝ x −1⎠
( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n ∀n ∈ 1
cos 2 θ = (1 + cos 2θ )
⎝ k =1 ⎠ 2 ⎛ 1 ± 1 − x2 ⎞
1 − cos 2θ sech −1 x = ln ⎜ ⎟, 0 < x ≤ 1
tg 2 θ = ⎜ x ⎟
⎝ ⎠
HIP
CO 1 + cos 2θ
⎛1 x2 + 1 ⎞
csch −1 x = ln ⎜ + ⎟, x ≠ 0
θ ⎜x x ⎟
⎝ ⎠
CA