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FORMULARIO DE
CÁLCULO DIFERENCIAL
VER.3.3
E INTEGRAL
Por Jesús Rubí M.
http://mx.geocities.com/estadisticapapers/

( a + b ) ⋅ ( a 2 − ab + b2 ) = a 3 + b3
( a + b ) ⋅ ( a 4 − a3b + a 2b 2 − ab3 + b 4 ) = a5 + b5

θ
0
30

n

k +1
( a + b ) ⋅  ∑ ( −1) a n −k b k −1  = a n + b n ∀2n − 1
 k =1

SUMAS Y PRODUCTOS

45

a1 + a2 +

VALOR ABSOLUTO

n

+ an = ∑ ak

a≤ a y −a ≤ a

k =1

ab = a b ó

k

+ bk ) = ∑ ak + ∑ bk

n

n

∏a
k =1

= c∑ ak

k =1

n

k

= ∏ ak

n

∑ (a

k =1

n

n

k =1
n

n

k

n

k =1

k =1

− ak −1 ) = an − a0

LOGARITMOS
log a N = x ⇒ a x = N

1+ 3 + 5 +

∑a
k =1

k

≤ ∑ ak
k =1

EXPONENTES
a p ⋅ a q = a p+q
ap
= a p−q
aq

(a )

pq

(a ⋅b)

=a
p

pq

= a p⋅bp

p

ap
a
=p
b
b
q

a p/q = a p

log a MN = log a M + log a N

log a N =



n



 k =1



y∈ −

-1

ππ
,
22

∀n ∈

arc ctg x
arc sec x
arc csc x

-2
-5

0

y ∈ 0, π

2

sen θ + cos 2 θ = 1
1 + ctg 2 θ = csc 2 θ

sen (θ + 2π ) = sen θ
cos (θ + 2π ) = cosθ
tg (θ + 2π ) = tg θ
sen (θ + π ) = − sen θ

-0.5

cos (θ + π ) = − cosθtg (θ + π ) = tg θ

-1
sen x
cos x
tg x

-1.5
-2
-8

-6

-4

-2

0

2

4

6

2.5

n

tg ( nπ ) = 0

-0.5
-1
-1.5
csc x
sec x
ctg x
-6

-4

-2

0

2

4

6

8

Gráfica 3. Las funciones trigonométricas inversas
arc sen x, arc cos x, arc tg x.

 2n + 1 
tg 
π=∞
2

π

sen θ = cos  θ − 
2


π

cosθ = sen  θ + 
2
sen (α ± β ) = sen α cos β ± cos α sen β

4

3

2

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

1

tg α ± tg β
tg (α ± β ) =
1 ∓ tg α tg β
sen 2θ = 2 sen θ cos θ

0

-1

arc sen x
arc cos x
arc tg x

CO
-2
-3

tg (θ + nπ ) = tg θ

sen ( nπ ) = 0

n
 2n + 1 
sen 
π  = ( −1)
2

 2n + 1 
cos 
π=0
2


0

π radianes=180

cos (θ + nπ ) = ( −1)cosθ
n

cos ( nπ ) = ( −1)

0.5

1
cscθ =
sen θ
1
secθ =
cos θ
1
ctg θ =
tg θ

sen (θ + nπ ) = ( −1) sen θ
n

8

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

-2.5
-8

-2

-1

0

1

5

IDENTIDADES TRIGONOMÉTRICAS

1

CA

senh x =

0

y ∈ [ 0, π ]

2

θ

FUNCIONES HIPERBÓLICAS

tg ( −θ ) = − tg θ

2

1

1.5

HIP

cos( −θ ) = cosθ

3

0

+ ( 2n − 1) = n 2

TRIGONOMETRÍA
CO
sen θ =
HIP
CA
cos θ =
HIP
sen θ CO
=
tg θ =
cos θ CA

tg 2 θ + 1 = sec 2 θ
sen ( −θ ) = − sen θ

4

1

-2

( a + b ) ⋅ ( a − b ) = a2 − b2
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 ) = a 3 + 3a 2b + 3ab 2 + b3
3
( a − b ) = a 3 − 3a 2b + 3ab 2 − b3
2
( a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
a − b ) ⋅ ( a 2 + ab + b 2 ) = a 3 − b 3
(
( a − b ) ⋅ ( a 3 + a 2b + ab 2 + b3 ) = a 4 − b 4
( a − b ) ⋅ ( a 4 + a 3b + a 2b 2 + ab3 + b 4 ) = a 5 − b5
( a − b ) ⋅  ∑ a n −kb k −1  = a n − b n



0

1
1
(α + β ) ⋅ cos (α − β )
2
2
1
1
sen α − sen β = 2 sen (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 sen (α + β ) ⋅ sen (α − β )
2
2
sen (α ± β )
tg α ± tg β =
cos α ⋅ cos β
1
sen α ⋅ cos β = 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
tg α + tg β
tg α ⋅ tg β =
ctg α + ctg β

3

1

k =1

a ⋅ ( c + d ) = ac + ad

2

0.5

n
n!
, k≤n
 =
 k  ( n − k )!k !
n
n  n− k k

n
( x + y) = ∑  x y
k =0  k 

log10 N = log N y log e N = ln N
ALGUNOS PRODUCTOS

3

Gráfica 4. Las funciones trigonométricas inversas
arc ctg x, arc...
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