Formulas integrales

Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3)

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( a + b ) ⋅ ( a 2 − ab + b 2 ) = a 3 + 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 ) = a 5 + b5 ( a + b ) ⋅ ( a5 − a 4 b + a 3b 2 − a 2 b3 + ab 4 − b5 ) = a 6 − b 6
⎛ n ⎞ k +1 ( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈ ⎝ k =1 ⎠impar par

Jesús Rubí M.

Fórmulas de Cálculo Diferencial e Integral VER.6.8
Jesús Rubí Miranda (jesusrubim@yahoo.com) http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO

θ 0 30 45 60 90

tg ctg cos sec csc sin 0 0 ∞ ∞ 1 1 12 3 2 3 2 3 2 1 3 1 2 1 2 2 2 1 1 3 1 3 2 2 3 3 2 12 0 0 ∞ ∞ 1 1

Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x :
4

sin α+ sin β = 2sin

3

2

y = ∠ sin x

⎧a si a ≥ 0 a =⎨ ⎩− a si a < 0 a = −a a ≤ a y −a ≤ a a ≥0 y a =0 ⇔ a=0 ab = a b ó
n n

( a + b ) ⋅ ⎜ ∑ ( −1)
a1 + a2 +

⎛

⎞ a n − k b k −1 ⎟ = a n − b n ∀ n ∈ ⎝ k =1 ⎠ SUMAS Y PRODUCTOS
n k +1

y ∈ ⎢− , ⎥ ⎣ 2 2⎦ y = ∠ cos x y ∈ [ 0, π ] y = ∠ tg x y∈ −

⎡ π π⎤

1

1 1 (α + β ) ⋅ cos (α − β ) 2 2 1 1 sin α − sin β = 2 sin (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 = −2 sin (α + β ) ⋅ sin (α − β ) 2 2 tg α ± tg β = sin (α ± β ) cos α ⋅ cos β

0

π π

-1

, 2 2 y ∈ 0, π
2

arc ctg x arc sec x arc csc x 0 5

-2 -5

+ an = ∑ ak
k =1

n

1 y = ∠ ctg x = ∠ tg x

IDENTIDADES TRIGONOMÉTRICAS
sin θ + cos 2 θ = 1 1 + ctg 2 θ = csc 2 θ tg 2 θ + 1 = sec 2 θ

∏ak =1

k n

= ∏ ak
k =1 k

∑ c = nc
k =1 n

n

a+b ≤ a + b ó

∑a
k =1

≤ ∑ ak
k =1

n

∑ ca
k =1 n k =1 n

k

= c ∑ ak
k =1 n n

n

1 y = ∠ sec x = ∠ cos y ∈ [ 0, π ] x 1 ⎡ π π⎤ y = ∠ csc x = ∠ sen y ∈ ⎢− , ⎥ x ⎣ 2 2⎦
k =1

1 ⎡sin (α − β ) + sin (α + β ) ⎤ ⎦ 2⎣ 1 sin α ⋅ sin β = ⎡cos (α − β ) − cos (α + β ) ⎤ ⎦ 2⎣ 1 cos α ⋅ cos β = ⎡cos (α − β ) + cos (α + β ) ⎤ ⎦2⎣ sin α ⋅ cos β = tg α ⋅ tg β = tg α + tg β ctg α + ctg β FUNCIONES HIPERBÓLICAS

sin ( −θ ) = − sin θ cos ( −θ ) = cos θ tg ( −θ ) = − tg θ
sin (θ + 2π ) = sin θ cos (θ + 2π ) = cos θ tg (θ + 2π ) = tg θ sin (θ + π ) = − sin θ cos (θ + π ) = − cos θ tg (θ + π ) = tg θ
sen x cos x tg x

∑ ( ak + bk ) = ∑ ak + ∑ bk
k =1

EXPONENTES

Gráfica 1. Las funciones trigonométricas: sin x , cosx , tg x :
2 1.5

a p ⋅ a q = a p+q ap = a p−q aq

∑(a
k =1

k

− ak −1 ) = an − a0

(a )

p q

=a
p

pq

(a ⋅b)
p

= a ⋅b
p

p

ap ⎛a⎞ ⎜ ⎟ = p b ⎝b⎠ a p/q = a p
q

LOGARITMOS log a N = x ⇒ a x = 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 log b N ln N = log a N = log b a ln a log10 N = log N y log e N = ln N
ALGUNOSPRODUCTOS a ⋅ ( c + d ) = ac + ad

n ∑ ⎡ a + ( k − 1) d ⎤ = 2 ⎡ 2a + ( n − 1) d ⎤ ⎣ ⎦ ⎣ ⎦ k =1 n = (a + l ) 2 n 1 − r n a − rl k −1 ∑ ar = a 1 − r = 1 − r k =1 n 1 ∑ k = 2 ( n2 + n ) k =1 n 1 ∑ k 2 = 6 ( 2n3 + 3n2 + n ) k =1 n 1 ∑ k 3 = 4 ( n 4 + 2n3 + n 2 ) k =1 n 1 ∑ k 4 = 30 ( 6n5 + 15n4 + 10n3 − n ) k =1 1+ 3 + 5 + n! = ∏ k
k =1 n

n

1

0.5

0 -0.5

-1

-1.5

-2 -8

sin (θ + nπ) = ( −1) sin θ
n
8

-6

-4

-2

0

2

4

6

Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x :
2.5 2 1.5 1 0.5

cos (θ + nπ ) = ( −1) cos θ
n

tg (θ + nπ ) = tg θ

ex − e− x 2 e x + e− x cosh x = 2 sinh x e x − e − x = tgh x = cosh x e x + e− x e x + e− x 1 = ctgh x = tgh x e x − e − x 1 2 = sech x = cosh x e x + e − x 1 2 = csch x = sinh x e x − e − x sinhx =
sinh : → → [1, ∞ → −1,1 − {0} → −∞ , −1 ∪ 1, ∞ → 0 ,1] − {0} → − {0} cosh : tgh : ctgh : sech : csch :

sin ( nπ ) = 0 cos ( nπ ) = ( −1) tg ( nπ ) = 0
n ⎛ 2n + 1 ⎞ sin ⎜ π ⎟ = ( −1) ⎝ 2 ⎠ ⎛ 2n + 1 ⎞ cos ⎜ π⎟=0 ⎝ 2 ⎠ ⎛ 2n + 1 ⎞ tg ⎜ π⎟=∞ ⎝ 2 ⎠ n

+ ( 2n − 1) = n

2
0 -0.5 -1 -1.5 -2 -2.5 -8 csc x sec x ctg x -6 -4 -2 0 2 4 6 8

Gráfica 5. Las funciones hiperbólicas sinh x ,...