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

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

( a + b ) ⋅ ( a 2 − ab + b 2 ) = a 3 + b 3
( a + b ) ⋅ ( a 3 − a 2 b + ab 2 − b 3 ) = a 4 − b 4
( a + b ) ⋅ ( a 4 − a 3 b + a 2 b 2 − ab 3 +b 4 ) = a 5 + b 5
( a + b ) ⋅ ( a 5 − a 4 b + a 3 b 2 − a 2 b 3 + ab 4 − b 5 ) = a 6 − b 6
⎛n

k +1
( a + b ) ⋅ ⎜ ∑ ( −1) a n− k b k −1 ⎟ = a n + b n ∀ n ∈
⎝ k =1

a n − k b k −1 ⎟ = a n − b n ∀ n ∈
⎝ k =1

SUMAS Y PRODUCTOS

a = −a

a1 + a2 +

a ≤ a y −a ≤ a
a ≥0 y a =0 ⇔ a=0
ab = a b ó

n

a+b ≤ a + b ó

k

n

n

≤ ∑ ak

k

k =1

(a ⋅b)

=a

nk =1

k =1
n

k =1

ap
= a p−q
aq
p

k =1

n

∑(a

k =1

= 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

1+ 3 + 5 +

log10 N = log N y log e N = ln N

ππ

,
22

tg 2 θ + 1 = sec 2 θsin ( −θ ) = − sin θ

sin θ + cos 2 θ = 1
2

1 + ctg 2 θ = csc 2 θ

( 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 2b + 3ab 2 + b33
( a − b ) = a 3 − 3a 2b + 3ab 2 − b3
2
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
2

( a − b ) ⋅ ( a + ab + b ) = a − b
( a − b ) ⋅ ( a 3 + a 2 b + ab 2 + b 3 ) = a 4 − b 4
( a − b ) ⋅ ( a 4 + a 3 b + a 2 b 2 + ab 3 + b 4 ) = a 5 − b 5
2

n

2

3

3

( a − b ) ⋅ ⎜ ∑ a n − k b k −1 ⎟ = a n − b n
⎝ k =1

∀n ∈

tg (θ + π ) = tg θ
s en x
co s x
tg x-1 . 5

-2

0

2

4

6

sin (θ + nπ ) = ( −1) sin θ
n

8

Gráfica 2. Las funciones trigonométricas csc x ,
sec x , ctg x :
2. 5

1
0. 5

2
0
-0 . 5
-1

k =1

( x1 + x2 +

+ xk ) = ∑
n

-1 . 5

-2

0

2

4

6

8

xknk

4

n
⎛ 2n + 1 ⎞
sin ⎜
π ⎟ = ( −1)
⎝2

⎛ 2n + 1 ⎞
cos ⎜
π⎟=0
⎝2

⎛ 2n + 1 ⎞
tg ⎜
π⎟=∞
⎝2

cosh :
tgh :
ctgh:

→ [1, ∞
→ −1 , 1
− {0} → −∞ , −1 ∪ 1, ∞

sech :

→ 0 ,1]

csch :

− {0} →

-1

ar c s en x
a r c co s x
ar c tg x
-2

-1

0

1

2

3

cos 2θ = cos 2 θ − sin 2 θ
2 tg θ
tg 2θ =
1 − tg 2 θ
1
sin 2 θ = (1 − cos 2θ )
2
1
cos 2 θ = (1 + cos 2θ )
2
1 − cos 2θ
tg 2 θ =
1 + cos 2θ

− {0}

Gráfica 5. Las funciones hiperbólicas sinh x ,
cosh x , tgh x :5
4

π⎞

sin θ = cos ⎜ θ − ⎟
2⎠

π⎞

cos θ = sin ⎜ θ + ⎟
2⎠

tg α ± tg β
tg (α ± β ) =
1 ∓ t g α tg β
sin 2θ = 2 sin θ cos θ

0

CO

sinh :
n

3
2
1
0
-1
-2

cos (α ± β ) = cos α cos β ∓ sin α sin β

1

-2
-3

tg (θ + nπ ) = tg θ

sin ( nπ ) = 0

sin (α ± β ) = sin α cos β ± cos α sin β

2

CA

-4

3

e = 2.71828182846…TRIGONOMETRÍA
CO
1
sen θ =
cscθ =
HIP
sen θ
CA
1
cosθ =
secθ =
HIP
cosθ
sen θ CO
1
tgθ =
ctgθ =
=
cosθ CA
tg θ

θ

-6

Gráfica 3. Las funciones trigonométricas inversas
arcsin x , arccos x , arctg x :

CONSTANTES
π = 3.14159265359…

HIP

cs c x
se c x
ctg x

-2

n!
n
x1n1 ⋅ x2 2
n1 ! n2 ! nk !

n

tg ( nπ ) = 0

1. 5

-2 . 5
-8

cos (θ + nπ )= ( −1) cos θ

cos ( nπ ) = ( −1)

2

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
sinh x =

cos (θ + π ) = − cos θ

-1

⎛n⎞
n!
, k≤n
⎜ ⎟=
⎝ k ⎠ ( n − k )!k !
n
⎛n⎞
n
( x + y ) = ∑ ⎜ ⎟ xn−k y k
k =0 ⎝ k ⎠...