Formulario Calculo

FORMULARIO DE
CÁLCULO DIFERENCIAL
VER.3.7
E INTEGRAL
Jesús Rubí Miranda (jesusrubi1@yahoo.com)
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

− ak −1 ) = an − a0

k −1k =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 − b 3
( 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


∞

2

3

2
2

2

1
1

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−k k
n
(x + y) = ∑  x y
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

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


sen (α ± β ) = sen α cos β ± cos α sen β

-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 =
coshx e x + e − x
1
2
csch x =
=
senh x e x − e − x
senh x =

cosh :
tgh :
ctgh :

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

sech :

→ 0 ,1]

csch :

− {0} →

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 = 2 sen 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

− {0}

Gráfica 5. Las funciones hiperbólicas senh x ,
cosh x , tgh x :
5

π

sen θ = cos  θ − 
2

π

cos θ = sen θ + 
2


1
0.5

n!
n
=∑
x1n1 ⋅ x2 2
n1 !n2 ! nk !

n

tg ( nπ ) = 0

sen x
cos x
tg x

-1.5

1
sen (α − β ) + sen (α + β ) 

2
1
sen α ⋅ sen β = cos (α − β ) − cos (α + β ) 

2
1cos α ⋅ cos β = cos (α − β ) + cos (α + β ) 

2
sen α ⋅ cos β =

senh :

sen ( nπ ) = 0

0

sen (α ± β )
cos α ⋅ cos β

FUNCIONES HIPERBÓLICAS

tg ( −θ ) = − tg θ

y ∈ 0, π

5

tg α ⋅ tg β =

n

n! = ∏ k

π radianes=180

2

sen ( −θ ) = − sen θ

1

+ ( 2n − 1) = n 2

+ xk )

IDENTIDADES TRIGONOMÉTRICAS
sen θ + cos 2 θ = 1
tg 2 θ + 1 = sec 2 θ

1.5...