1. Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
Fórmulas de ( a + b ) ⋅ ( a − ab + b ) = a + b
2 2 3 3 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
2 2
Cálculo Diferencial HIP
CO 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 ACTUALIZADO AGO-2007 θ 3
2 2
( a + b ) ⋅ ( a5 − a 4 b + a 3b 2 − a 2 b3 + ab 4 − b5 ) = a 6 − b 6
1 1
CA cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β )
Jesús Rubí Miranda (jesusrubim@yahoo.com) 2 2 2
Móvil. Méx. DF. 044 55 13 78 51 94 1 1
⎛ ⎞ cos α − cos β = −2 sin (α + β ) ⋅ sin (α − β )
n
( a + b ) ⋅ ⎜ ∑ ( −1)
k +1
a n − k b k −1 ⎟ = a n + b n ∀ n ∈ impar θ sin cos tg ctg sec csc
1
2 2
⎝ k =1 ⎠
1. VALOR ABSOLUTO 0 0 1 0 ∞ 1 ∞
sin (α ± β )
0
⎛ n
⎞ tg α ± tg β =
( a + b ) ⋅ ⎜ ∑ ( −1)
k +1 12
⎧a si a ≥ 0 a n − k b k −1 ⎟ = a n − b n ∀ n ∈ par 30 3 2 1 3 3 2 3 2
a =⎨ ⎝ k =1 ⎠
-1
arc ctg x cos α ⋅ cos β
45 1 2 1 2 2 2
⎩− a si a < 0
1 1 arc sec x
arc csc x
1
⎡sin (α − β ) + sin (α + β ) ⎤
5. SUMAS Y PRODUCTOS 60 3 2 12 3 1 3 2 2 3 -2
sin α ⋅ cos β =
a = −a 2⎣ ⎦
-5 0 5
∞ ∞
n
+ a n = ∑ ak
90 1 0 0 1
a ≤ a y −a ≤ a a1 + a2 + 8. IDENTIDADES TRIGONOMÉTRICAS 1
k =1 ⎡ π π⎤ 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 y = ∠ sin x y ∈ ⎢− , ⎥
∑ c = nc ⎣ 2 2⎦
1 + ctg 2 θ = csc 2 θ
cos α ⋅ cos β =
1
⎡cos (α − β ) + cos (α + β ) ⎤
y = ∠ cos x y ∈ [ 0, π ]
n n
∏a = ∏ ak 2⎣ ⎦
k =1
ab = a b ó k n n tg 2 θ + 1 = sec 2 θ
k =1 k =1
∑ ca = c ∑ ak
y = ∠ tg x y∈ −
π π tg α + tg β
sin ( −θ ) = − sin θ
k
n n k =1 k =1
, tg α ⋅ tg β =
a+b ≤ a + b ó ∑a ≤ ∑ ak n n n
2 2 ctg α + ctg β
cos ( −θ ) = cos θ
k
k =1 k =1
∑(a k + bk ) = ∑ ak + ∑ bk
y = ∠ ctg x = ∠ tg
1
y ∈ 0, π 9. FUNCIONES HIPERBÓLICAS
tg ( −θ ) = − tg θ
k =1 k =1 k =1
2. EXPONENTES x ex − e− x
sinh x =
n
a p ⋅ a q = a p+q ∑(a
k =1
k − ak −1 ) = an − a0 y = ∠ sec x = ∠ cos
1
y ∈ [ 0, π ] sin (θ + 2π ) = sin θ 2
ap x
= a p−q e x + e− x
n
n 1 ⎡ π π⎤ cos (θ + 2π ) = cos θ cosh x =
aq
∑ ⎡ a + ( k − 1) d ⎤ = 2 ⎡ 2a + ( n − 1) d ⎤
⎣ ⎦ ⎣ ⎦ y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥
⎣ 2 2⎦ tg (θ + 2π ) = tg θ
2
(a )
p q
=a pq k =1 x
tgh x =
sinh x e x − e − x
=
n
(a + l ) Gráfica 1. Las funciones trigonométricas: sin x , sin (θ + π ) = − sin θ cosh x e x + e− x
(a ⋅b) = a ⋅b =
p p p
2 cos x , tg x : cos (θ + π ) = − cos θ 1 e x + e− x
p n
1 − r n a − rl ctgh x = =
⎛a⎞ ap
⎜ ⎟ = p ∑ ar k −1 = a 1 − r = 1 − r 2 tg (θ + π ) = tg θ tgh x e x − e − x
⎝b⎠ b k =1
1 2
sin (θ + nπ ) = ( −1) sin θ sech x = =
1.5 n
∑ k = 2 ( n2 + n )
n
a = a 1
cosh x e x + e − x
p/q q p
1
cos (θ + nπ ) = ( −1) cos θ
n
3. LOGARITMOS k =1 1 2
n ( n + 1)( 2n + 1)
0.5
csch x = =
∑ k 2 = 6 ( 2n3 + 3n2 + n ) = tg (θ + nπ ) = tg θ sinh x e x − e − x
n
log a N = x ⇒ a x = N 1
0
log a MN = log a M + log a N k =1 6 sinh : →
-0.5
sin ( nπ ) = 0
∑ k = 4 ( n + 2n + n ) → [1, ∞
n
M 3 1 4 3 2 cosh :
= log a M − log a N cos ( nπ ) = ( −1)
-1
n
log a k =1
N -1.5 sen x
tgh : → −1,1
tg ( nπ ) = 0
cos x
∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n )
n
log a N = r log a N
r 1
− {0} → −∞ , −1 ∪ 1, ∞
tg x
-2
-8 -6 -4 -2 0 2 4 6 8 ctgh :
k =1
⎛ 2n + 1 ⎞
π ⎟ = ( −1)
log b N ln N
→ 0 ,1]
n
log a N = = 1+ 3 + 5 + + ( 2n − 1) = n 2 Gráfica 2. Las funciones trigonométricas csc x , sin ⎜ sech :
log b a ln a ⎝ 2 ⎠
sec x , ctg x : csch : − {0} → − {0}
⎛ 2n + 1 ⎞
n
log10 N = log N y log e N = ln N n! = ∏ k cos ⎜ π⎟=0
4. ALGUNOS PRODUCTOS k =1 2.5 ⎝ 2 ⎠ Gráfica 5. Las funciones hiperbólicas sinh x ,
a ⋅ ( c + d ) = ac + ad ⎛n⎞ n! 2
⎛ 2n + 1 ⎞ cosh x , tgh x :
⎜ ⎟= , k≤n tg ⎜ π⎟=∞
⎝ k ⎠ ( n − k ) !k !
1.5
( a + b ) ⋅ ( a − b ) = a 2 − b2 1
⎝ 2 ⎠ 5
4
n
⎛n⎞ ⎛ π⎞
( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2 ( x + y ) = ∑ ⎜ ⎟ x n −k y k
2 n 0.5
sin θ = cos ⎜ θ − ⎟ 3
k =0 ⎝ k ⎠ ⎝ 2⎠
0
( a − b ) ⋅ ( a − b ) = ( a − b ) = a 2 − 2ab + b 2
2
2
-0.5
⎛ π⎞ 1
( x1 + x2 + + xk ) = ∑
n! cos θ = sin ⎜ θ + ⎟
( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd x1n1 ⋅ x2 2
n -1
n
xknk
⎝ 2⎠ 0
n1 ! n2 ! nk ! -1.5
( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd
csc x -1
sin (α ± β ) = sin α cos β ± cos α sin β
-2 sec x
6. CONSTANTES ctg x -2
( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd
-2.5
π = 3.14159265359… cos (α ± β ) = cos α cos β ∓ sin α sin β
-8 -6 -4 -2 0 2 4 6 8 senh x
-3 cosh x
tgh x
( a + b ) = a3 + 3a 2b + 3ab 2 + b3 e = 2.71828182846…
3 -4
Gráfica 3. Las funciones trigonométricas inversas tg α ± tg β -5 0 5
tg (α ± β ) = 10. FUNCIONES HIPERBÓLICAS INV
( a − b ) = a 3 − 3a 2b + 3ab 2 − b3
3 7. TRIGONOMETRÍA arcsin x , arccos x , arctg x : 1 ∓ tg α tg β
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
2 sen θ =
CO
cscθ =
1 4 sin 2θ = 2sin θ cos θ (
sinh −1 x = ln x + x 2 + 1 , ∀x ∈ )
sen θ
( )
HIP
cos 2θ = cos θ − sin θ
2 2
cosh −1 x = ln x ± x 2 − 1 , x ≥ 1
( a − b ) ⋅ ( a 2 + ab + b 2 ) = a 3 − b3
3
CA 1
cosθ = secθ = 2 tg θ
HIP cosθ tg 2θ = 1 ⎛1+ x ⎞
( a − b ) ⋅ ( a 3 + a 2 b + ab 2 + b3 ) = a 4 − b 4 1 − tg 2 θ
2
sen θ CO 1 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
1 ⎛ x +1⎞
0
2 ctgh −1 x = ln ⎜ ⎟, x >1
⎛ n
⎞ 2 ⎝ x −1⎠
( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n ∀n ∈ π radianes=180 1
cos 2 θ = (1 + cos 2θ )
⎝ k =1 ⎠ -1
arc sen x
2 ⎛ 1 ± 1 − x2 ⎞
sech −1 x = ln ⎜ ⎟, 0 < x ≤ 1
arc cos x
arc tg x
1 − cos 2θ ⎜ ⎟
-2
-3 -2 -1 0 1 2 3 tg 2 θ = ⎝ x ⎠
1 + cos 2θ
⎛1 x2 + 1 ⎞
csch −1 x = ln ⎜ + ⎟, x ≠ 0
⎜x x ⎟
⎝ ⎠