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
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 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
