FORMULARIO DE CÁLCULO DIFERENCIAL VER.3.6 E INTEGRAL Jesús Rubí Miranda ([email protected]) http://mx.geocities.com/estadisticapapers/ http://mx.geocities.com/dicalculus/ VALOR ABSOLUTO a si a ≥ 0 a = −a si a < 0 a = −a n = ∏ ak k k =1 n n ∑a k =1 ≤ ∑ ak k k =1 EXPONENTES a ⋅a = a p q ( a ⋅ b) k =1 0 12 0 1 3 2 1 3 30 1 n k =1 k =1 n n n ∑ ar k −1 k =1 n (a + l ) 2 n 1− r a − rl =a = 1− r 1− r n 1 ∑ k = 2 (n q ap/q = ap k =1 LOGARITMOS n log a MN = log a M + log a N M = log a M − log a N N log a N r = r log a N log a logb N ln N = logb a ln a 2 ALGUNOS PRODUCTOS a ⋅ ( c + d ) = ac + ad (a + b) ⋅ ( a − b) = a − b 2 ( a + b ) ⋅ ( a + b ) = ( a + b ) = a 2 + 2ab + b 2 2 − ⋅ − = − a b a b a b ( )( ) ( ) = a 2 − 2ab + b 2 ( x + b ) ⋅ ( x + d ) = x 2 + ( b + d ) x + bd ( ax + b ) ⋅ ( cx + d ) = acx 2 + ( ad + bc ) x + bd ( a + b ) ⋅ ( c + d ) = ac + ad + bc + bd 3 ( a + b ) = a3 + 3a 2 b + 3ab 2 + b3 3 ( a − b ) = a 3 − 3a 2 b + 3ab 2 − b3 2 ( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc 2 ( a − b ) ⋅ ( a 2 + ab + b 2 ) = a 3 − b3 ( a − b ) ⋅ ( a3 + a 2 b + ab2 + b3 ) = a 4 − b 4 ( a − b ) ⋅ ( a 4 + a 3b + a 2 b 2 + ab3 + b 4 ) = a 5 − b5 n ( a − b ) ⋅ ∑ a n −k b k −1 = a n − b n ∀n ∈ k =1 0 ∞ y = ∠ cos x y = ∠ tg x y∈ − 1 x ∞ 3 sec csc 1 ∞ 2 3 2 2 0 ∞ 1 cos ( −θ ) = cos θ sen (θ + 2π ) = sen θ cos (θ + 2π ) = cosθ tg (θ + 2π ) = tg θ sen (θ + π ) = − sen θ cos (θ + π ) = − cosθ tg (θ + π ) = tg θ sen (θ + nπ ) = ( −1) sen θ n cos (θ + nπ ) = ( −1) cos θ tg (θ + nπ ) = tg θ 0.5 -0.5 cos ( nπ ) = ( −1) -1 -2 -8 -6 -4 -2 0 2 4 6 8 Gráfica 2. Las funciones trigonométricas csc x , n sec x , ctg x : k =1 n n! , k≤n = k ( n − k )! k ! n n n ( x + y ) = ∑ x n−k y k k =0 k n 2.5 2 1.5 CONSTANTES π = 3.14159265359… e = 2.71828182846… TRIGONOMETRÍA CO sen θ = HIP CA cosθ = HIP sen θ CO tg θ = = cos θ CA 0 nk k x -0.5 -1 sen (α ± β ) = sen α cos β ± cos α sen β -1.5 csc x sec x ctg x -2 -2.5 -8 1 sen θ 1 secθ = cos θ 1 ctg θ = tg θ cscθ = -6 -4 -2 0 2 4 6 8 Gráfica 3. Las funciones trigonométricas inversas arcsen x , arccos x , arctg x : 4 3 2 1 0 -1 -2 -3 n 2n + 1 sen π = ( −1) 2 2n + 1 cos π=0 2 2n + 1 tg π=∞ 2 π sen θ = cos θ − 2 π cos θ = sen θ + 2 1 0.5 n! =∑ x1n1 ⋅ x2n2 n1 !n2 ! nk ! n tg ( nπ ) = 0 sen x cos x tg x -1.5 arc sen x arc cos x arc tg x -2 -1 0 1 2 3 tg α + tg β ctg α + ctg β e x − e− x 2 e x + e− x cosh x = 2 senh x e x − e − x tgh x = = cosh x e x + e− x 1 e x + e− x = ctgh x = tgh x e x − e − x 1 2 = sech x = cosh x e x + e − x 1 2 csch x = = senh x e x − e − x senh x = senh : sen ( nπ ) = 0 0 1 sen (α − β ) + sen (α + β ) 2 1 sen α ⋅ sen β = cos (α − β ) − cos (α + β ) 2 1 cos α ⋅ cos β = cos (α − β ) + cos (α + β ) 2 FUNCIONES HIPERBÓLICAS tg ( −θ ) = − tg θ y ∈ 0, π sen (α ± β ) cos α ⋅ cos β sen α ⋅ cos β = tg α ⋅ tg β = n n! = ∏ k π radianes=180 2 sen ( −θ ) = − sen θ π π , 2 2 5 IDENTIDADES TRIGONOMÉTRICAS sen θ + cos 2 θ = 1 tg 2 θ + 1 = sec 2 θ 1 + ( 2n − 1) = n 2 + xk ) 0 1 + ctg 2 θ = csc2 θ 1.5 = ( x1 + x2 + tg α ± tg β = arc ctg x arc sec x arc csc x -2 -5 3 2 + n) 1 1 (α + β ) ⋅ cos (α − β ) 2 2 1 1 sen α − sen β = 2sen (α − β ) ⋅ cos (α + β ) 2 2 1 1 cos α + cos β = 2 cos (α + β ) ⋅ cos (α − β ) 2 2 1 1 cos α − cos β = −2sen (α + β ) ⋅ sen (α − β ) 2 2 sen α + sen β = 2 sen 0 -1 2 2 2 1 1 3 k =1 1+ 3 + 5 + log10 N = log N y log e N = ln N 2 1 ( 2n3 + 3n2 + n ) 6 n 1 k 3 = ( n 4 + 2n3 + n 2 ) ∑ 4 k =1 n 1 k 4 = ( 6n5 + 15n4 + 10n3 − n ) ∑ 30 k =1 ∑k log a N = x ⇒ a x = N 1 3 ctg 1 y = ∠ sec x = ∠ cos y ∈ [ 0, π ] x 1 π π y = ∠ csc x = ∠ sen y ∈ − , x 2 2 Gráfica 1. Las funciones trigonométricas: sen x , cos x , tg x : = p 2 12 y = ∠ ctg x = ∠ tg − ak −1 ) = an − a0 k tg π π y ∈ − , 2 2 y ∈ [ 0, π ] y = ∠ sen x k =1 ap a = p b b 2 1 1 ∑ a + ( k − 1) d = 2 2a + ( n − 1) d 2 0 90 + bk ) = ∑ ak + ∑ bk k = ap ⋅ bp log a N = 1 cos k =1 n 2 sen k =1 n ∑(a 3 θ 3 2 = c ∑ ak k k =1 4 CA 45 n ∑ ca ∑(a = a pq p n CO θ 60 k =1 n a = a p −q aq p q HIP n ∑ c = nc k =1 p+q p (a ) par Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : n n ∏a a+b ≤ a + b ó impar + a n = ∑ ak a1 + a2 + a ≥0y a =0 ⇔ a=0 k =1 n k +1 ( a + b ) ⋅ ∑ ( −1) a n− k b k −1 = a n + b n ∀ n ∈ k =1 n k +1 ( a + b ) ⋅ ∑ ( −1) a n− k b k −1 = a n − b n ∀ n ∈ k =1 SUMAS Y PRODUCTOS a≤ a y −a≤ a ab = a b ó ( a + b ) ⋅ ( a 2 − ab + b 2 ) = a3 + b3 ( a + b ) ⋅ ( a3 − a 2 b + ab 2 − b3 ) = a 4 − b 4 ( a + b ) ⋅ ( a 4 − a 3b + a 2 b 2 − ab3 + b 4 ) = a5 + b5 ( a + b ) ⋅ ( a5 − a 4 b + a3b 2 − a 2 b3 + ab4 − b5 ) = a 6 − b6 cos (α ± β ) = cos α cos β ∓ sen α sen β tg α ± tg β tg (α ± β ) = 1 ∓ tg α tg β sen 2θ = 2sen θ cosθ cos 2θ = cos 2 θ − sen 2 θ 2 tg θ tg 2θ = 1 − tg 2 θ 1 sen 2 θ = (1 − cos 2θ ) 2 1 cos 2 θ = (1 + cos 2θ ) 2 1 − cos 2θ tg 2 θ = 1 + cos 2θ cosh : tgh : ctgh : → → [1, ∞ → −1,1 − {0} → −∞ , −1 ∪ 1, ∞ sech : → 0,1] csch : − {0} → − {0} Gráfica 5. Las funciones hiperbólicas senh x , cosh x , tgh x : 5 4 3 2 1 0 -1 -2 senh x cosh x tgh x -3 -4 -5 0 5 FUNCS HIPERBÓLICAS INVERSAS ( ( ) ) senh −1 x = ln x + x 2 + 1 , ∀x ∈ cosh −1 x = ln x ± x 2 − 1 , x ≥ 1 tgh −1 x = 1 1+ x ln , 2 1− x ctgh −1 x = 1 x +1 ln , 2 x −1 x <1 x >1 1 ± 1 − x2 , 0 < x ≤ 1 sech −1 x = ln x 2 1 x +1 −1 , x ≠ 0 csch x = ln + x x IDENTIDADES DE FUNCS HIP cosh 2 x − senh 2 x = 1 1 − tgh 2 x = sech 2 x ctgh 2 x − 1 = csch x senh ( − x ) = − senh x cosh ( − x ) = cosh x tgh ( − x ) = − tgh x senh ( x ± y ) = senh x cosh y ± cosh x senh y cosh ( x ± y ) = cosh x cosh y ± senh x senh y tgh x ± tgh y 1 ± tgh x tgh y senh 2 x = 2senh x cosh x tgh ( x ± y ) = cosh 2 x = cosh 2 x + senh 2 x 2 tgh x tgh 2 x = 1 + tgh 2 x 1 ( cosh 2 x − 1) 2 1 cosh 2 x = ( cosh 2 x + 1) 2 cosh 2 x − 1 tgh 2 x = cosh 2 x + 1 senh 2 x = senh 2 x cosh 2 x + 1 OTRAS tgh x = ax + bx + c = 0 2 −b ± b 2 − 4ac 2a b 2 − 4ac = discriminante ⇒ x= LÍMITES 1 lim (1 + x ) x = e = 2.71828... x→0 x 1 lim 1 + = e x →∞ x sen x =1 lim x→0 x 1 − cos x lim =0 x→0 x ex − 1 lim =1 x→0 x x −1 lim =1 x →1 ln x DERIVADAS Dx f ( x ) = f ( x + ∆x ) − f ( x ) df ∆y = lim = lim ∆x → 0 ∆x dx ∆x →0 ∆x d (c) = 0 dx d ( cx ) = c dx d ( cx n ) = ncx n−1 dx d du dv dw (u ± v ± w ± ) = ± ± ± dx dx dx dx d du ( cu ) = c dx dx d dv du ( uv ) = u + v dx dx dx d dw dv du ( uvw) = uv + uw + vw dx dx dx dx d u v ( du dx ) − u ( dv dx ) = dx v v2 d n n −1 du u = nu ( ) dx dx dF dF du (Regla de la Cadena) = ⋅ dx du dx du 1 = dx dx du dF dF du = dx dx du x = f1 ( t ) dy dy dt f 2′ ( t ) = = donde dx dx dt f1′( t ) y = f 2 ( t ) DERIVADA DE FUNCS LOG & EXP du dx 1 du d = ⋅ ( ln u ) = dx u u dx d log e du ⋅ ( log u ) = dx u dx log e du d ( log a u ) = a ⋅ a > 0, a ≠ 1 dx u dx d u du e ) = eu ⋅ ( dx dx d u du a ) = a u ln a ⋅ ( dx dx d v du dv + ln u ⋅ u v ⋅ u ) = vu v −1 ( dx dx dx DERIVADA DE FUNCIONES TRIGO d du ( sen u ) = cos u dx dx d du ( cos u ) = − sen u dx dx d du ( tg u ) = sec2 u dx dx d du ( ctg u ) = − csc2 u dx dx d du ( sec u ) = sec u tg u dx dx d du ( csc u ) = − csc u ctg u dx dx d du ( vers u ) = sen u dx dx DERIV DE FUNCS TRIGO INVER 1 d du ⋅ ( ∠ sen u ) = dx 1 − u 2 dx 1 d du ⋅ ( ∠ cos u ) = − dx 1 − u 2 dx 1 d du ⋅ ( ∠ tg u ) = dx 1 + u 2 dx 1 d du ⋅ ( ∠ ctg u ) = − dx 1 + u 2 dx 1 d du + si u > 1 ⋅ ( ∠ sec u ) = ± dx u u 2 − 1 dx − si u < −1 1 d du − si u > 1 ⋅ ( ∠ csc u ) = ∓ dx u u 2 − 1 dx + si u < −1 du 1 d ⋅ ( ∠ vers u ) = dx 2u − u 2 dx DERIVADA DE FUNCS HIPERBÓLICAS d du senh u = cosh u dx dx d du cosh u = senh u dx dx d du tgh u = sech 2 u dx dx d du ctgh u = − csch 2 u dx dx d du sech u = − sech u tgh u dx dx d du csch u = − csch u ctgh u dx dx DERIVADA DE FUNCS HIP INV d 1 du senh −1 u = ⋅ dx 1 + u 2 dx -1 + d ±1 du si cosh u > 0 cosh −1 u = ⋅ , u >1 -1 dx u 2 − 1 dx − si cosh u < 0 d 1 du ⋅ , u <1 tgh −1 u = dx 1 − u 2 dx d 1 du ⋅ , u >1 ctgh −1 u = dx 1 − u 2 dx −1 ∓1 d du − si sech u > 0, u ∈ 0,1 ⋅ sech −1 u = −1 dx u 1 − u 2 dx + si sech u < 0, u ∈ 0,1 d 1 du csch −1 u = − ⋅ , u≠0 dx u 1 + u 2 dx INTEGRALES DEFINIDAS, PROPIEDADES ∫ ∫ ∫ ∫ ∫ b a b a { f ( x ) ± g ( x )} dx = ∫ f ( x ) dx ± ∫ g ( x ) dx b b a a b cf ( x ) dx = c ⋅ ∫ f ( x ) dx b a b a a a c∈ a c b f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx a c a f ( x ) dx = − ∫ f ( x ) dx b f ( x ) dx = 0 b m ⋅ ( b − a ) ≤ ∫ f ( x ) dx ≤ M ⋅ ( b − a ) a ⇔ m ≤ f ( x ) ≤ M ∀x ∈ [ a, b ] , m, M ∈ b b a a ∫ f ( x ) dx ≤ ∫ g ( x ) dx ⇔ f ( x ) ≤ g ( x ) ∀x ∈ [ a , b ] b b a a ∫ f ( x ) dx ≤ ∫ f ( x ) dx si a < b INTEGRALES ∫ adx =ax ∫ af ( x ) dx = a ∫ f ( x ) dx ∫ ( u ± v ± w ± ) dx = ∫ udx ± ∫ vdx ± ∫ wdx ± ∫ udv = uv − ∫ vdu ( Integración por partes ) ∫u n du = u n +1 n ≠ −1 n +1 du ∫ u = ln u INTEGRALES DE FUNCS LOG & EXP ∫ e du = e u u a u a > 0 ∫ a du = ln a a ≠ 1 u au −1 1 ∫ ua du = ln a ⋅ u − ln a u 1 = ln tgh u 2 ∫ ue du = e ( u − 1) ∫ ln udu =u ln u − u = u ( ln u − 1) u ∫ tgh udu = ln cosh u ∫ ctgh udu = ln senh u ∫ sech udu = ∠ tg ( senh u ) ∫ csch udu = − ctgh ( cosh u ) u INTREGRALES DE FRAC 1 u ∫ log a udu =ln a ( u ln u − u ) = ln a ( ln u − 1) u2 ∫ u log a udu = 4 ⋅ ( 2 log a u − 1) u2 ∫ u ln udu = 4 ( 2 ln u − 1) INTEGRALES DE FUNCS TRIGO ∫ sen udu = − cos u ∫ cos udu = sen u ∫ sec udu = tg u ∫ csc udu = − ctg u ∫ sec u tg udu = sec u ∫ csc u ctg udu = − csc u ∫ tg udu = − ln cos u = ln sec u ∫ ctg udu = ln sen u ∫ sec udu = ln sec u + tg u ∫ csc udu = ln csc u − ctg u 1 du u = ∠ tg a + a2 a 1 u = − ∠ ctg a a du 1 u−a ∫ u 2 − a 2 = 2a ln u + a du 1 a+u ∫ a 2 − u 2 = 2a ln a − u ∫u 2 du ∫ = ∠ sen a2 − u2 2 ( du ∫ ∫e ∫ u sen udu = sen u − u cos u du a2 ± u 2 du u a ) = au sen bu du = au ∫ e cos bu du = ∫ u cos udu = cos u + u sen u ) e au ( a sen bu − b cos bu ) a2 + b2 e au ( a cos bu + b sen bu ) a2 + b2 ALGUNAS SERIES INTEGRALES DE FUNCS TRIGO INV ∫ ∠ sen udu = u∠ sen u + 1 − u ∫ ∠ cos udu = u∠ cos u − 1 − u ∫ ∠ tg udu = u∠ tg u − ln 1 + u ∫ ∠ ctg udu = u∠ ctg u + ln 1 + u ∫ ∠ sec udu = u∠ sec u − ln ( u + u 2 2 + + f ( n ) ( x0 )( x − x0 ) n! f ( x ) = f ( 0) + f '( 0) x + 2 2 −1 = u∠ sec u − ∠ cosh u ∫ ∠ csc udu = u∠ csc u + ln ( u + u2 − 1 = u∠ csc u + ∠ cosh u INTEGRALES DE FUNCS HIP ) ) f '' ( x0 )( x − x0 ) f ( x ) = f ( x0 ) + f ' ( x0 )( x − x0 ) + 2 2 < a2 ) ( udu = − ( ctg u + u ) 2 2 1 u ln a a + a2 ± u 2 1 a ∫ u u 2 − a 2 = a ∠ cos u 1 u = ∠ sec a a u 2 a2 u 2 2 2 ∫ a − u du = 2 a − u + 2 ∠ sen a 2 u 2 a 2 2 2 2 2 ∫ u ± a du = 2 u ± a ± 2 ln u + u ± a MAS INTEGRALES udu = ∫ senh udu = cosh u ∫ cosh udu = senh u ∫ sech udu = tgh u ∫ csch udu = − ctgh u ∫ sech u tgh udu = − sech u ∫ csch u ctgh udu = − csch u (u = ln u + u 2 ± a 2 u 2 ± a2 ∫u u 1 − sen 2u 2 4 u 1 2 ∫ cos udu = 2 + 4 sen 2u 2 ∫ tg udu = tg u − u ∫ ctg > a2 ) u a = −∠ cos 2 2 2 INTEGRALES CON 2 ∫ sen (u + + f ( n) ( 0) x 2! n f '' ( 0 ) x : Taylor 2 2! n : Maclaurin n! x 2 x3 xn + + + + 2! 3! n! 3 5 7 x x x x 2 n −1 n −1 sen x = x − + − + + ( −1) 3! 5! 7! ( 2n − 1)! ex = 1 + x + cos x = 1 − x2 x4 x6 + − + 2! 4! 6! + ( −1) n −1 x 2n−2 ( 2n − 2 ) ! n x 2 x3 x 4 n −1 x + − + + ( −1) 2 3 4 n 2 n −1 x3 x5 x7 n −1 x ∠ tg x = x − + − + + ( −1) 3 5 7 2n − 1 ln (1 + x ) = x − 2