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# formulas calculo integral y diferencial

## on Jan 21, 2012

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aqui les dejo un formulario que en lo personal me ha servido mucho..

aqui les dejo un formulario que en lo personal me ha servido mucho..

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## formulas calculo integral y diferencialPresentation Transcript

• 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 2Cálculo 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 + b5e 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 ∞ 1http://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⎤ 1http://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 TRIGONOMÉTRICASa ≤ 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− xa 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 xa 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) nlog a k =1 2 N tgh : → −1,1 ∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n ) n 1 tg ( nπ ) = 0 1.5log 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 xa ⋅ ( 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 ⎟ ⎝ ⎠
• 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 + vcosh 2 x − senh 2 x = 1 d du ∫ e du = e u u dx dx dx senh u = cosh u1 − 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 uctgh x − 1 = csch x 2 ⎩ d ⎛ u ⎞ v ( du dx ) − u ( dv dx ) dx dxsenh ( − 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 ⎟ ucosh ( − x ) = cosh x dx dx ⎝ ⎠ 1 = ln tgh u d n ( u ) = nu n−1 dutgh ( − x ) = − tgh x d du ∫ ue du = e ( u − 1) 2 u u dx dx ctgh u = − csch 2 u dx dx INTEGRALES DE FRACsenh ( 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 ucosh ( 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 utgh ( 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−asenh 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 u2cosh 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 2tgh 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 CONsenh 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 acosh 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 atgh 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 2tgh 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 = lne 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 2e − 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 bax + 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 aexp (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 ) 1lim (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 alim ALGUNAS SERIESx →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 blim ∫ ∠ sen udu = u∠ sen u + 1 − u 2! 2x →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 + + 2lim : Taylor ∫ adx =axx →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 dulim 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 + + : MaclaurinDx 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 −1d ( 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 −1ddx ( 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 nd 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 udx dx dx dx 2 2 n −1d du d 1 du x3 x5 x7 ( cu ) = c ( ∠ vers u ) = ⋅ ∠ tg x = x − + − + + ( −1) n −1 x ∫ csch udu = − ctgh u 2dx dx dx 2u − u 2 dx 3 5 7 2n − 1 ∫ sech u tgh udu = − sech u ∫ csch u ctgh udu = − csch u
• Formulario de Cálculo Diferencial e Integral Jesús Rubí M. ÁLGEBRA LINEALDef. El determinante de una matriz ⎡a a ⎤ A = ⎢ 11 12 ⎥ ⎣ a21 a22 ⎦está dado por a11 a12 det A = = a11a22 − a12 a21 . a21 a22Def. 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 ⋅ a31det A = a21 a22 a23 = + a13 ⋅ a21 ⋅ a32 − a11 ⋅ a23 ⋅ a32 . a31 a32 a33 −a12 ⋅ a21 ⋅ a33 − a13 ⋅ a22 ⋅ a31