Formulas De Calculo

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Formulas De Calculo

  1. 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 ⎟ ⎝ ⎠

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