Ing Civil

Formulario de Cálculo Diferencial e Integral

Formulario de
Cálculo Diferencial
e Integral
VER.4.3
Jesús Rubí Miranda (jesusrubim@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

∏a

a+b ≤ a + b ó

n

∑ ca

k =1

n

k =1

k =1

n

k

≤ ∑ ak

∑(a

k =1

∑(acsc

1
1
∞
∞
323 2
321 3
2
2
1
1
1 21 2
313
2 23
3 2 12
0
0
1
1
∞
∞
0
12

y = ∠ sen x
y = ∠ cos x

⎡ π π⎤
y ∈ ⎢− , ⎥
⎣ 2 2⎦
y ∈ [ 0, π ]
y∈ −

k =1

n

k =1

k =1

k

+ bk ) = ∑ ak + ∑ bk

k

− ak −1 ) = an − a0

n

∑ ar

k −1

k =1

,
22

log a MN = log a M + log a N
M
log a
= log a M − log a N
N
log a N r = r log a N
log b Nln N
=
log b a ln a

cos (θ + π ) = − cosθ

2

( a − b ) ⋅ ( a + ab + b ) = a − b
( a − b ) ⋅ ( a3 + a 2 b + ab2 + b3 ) = a 4 − b4
( a − b ) ⋅ ( a 4 + a3b + a 2 b2 + ab3 + b 4 ) = a5 − b5
2

n

2

⎞

3

3

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

⎠

∀n ∈

-4

-2

0

2

4

6

n

8

2. 5

0. 5
0
-0 . 5
-1
-1 . 5

+ xk )

cs c x
sec x
ct g x

-2
-2 . 5
-8

-6

-4

-2

0

2

4

6

8

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

n!
n
=∑
x1n1 ⋅ x2 2
n1 !n2 ! nk !

nk
k

x

θ
CA

π⎞
⎛
sen θ = cos ⎜θ − ⎟
2⎠
⎝

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

1

tg α ± tg β
tg (α ± β ) =
1 ∓ tg α tg β
sen2θ = 2 sen θ cosθ

-1

-2
-3

π radianes=180

HIP

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

2

0

CO

ar c s en x
a r c co s x
ar c t g x
-2

-1

0

1

2

cos 2θ = cos 2 θ − sen 2 θ
3

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 :
n

π⎞
⎛
cosθ = sen ⎜θ + ⎟
2⎠
⎝

4

3

CONSTANTES
π = 3.14159265359…
e = 2.71828182846…
TRIGONOMETRÍA
CO
1
sen θ =
cscθ =
HIP
sen θ
CA
1
cosθ =
secθ =
HIP
cosθ
sen θ CO
1
tg θ =
ctg θ =
=
cosθ CA
tg θ

tg (θ + nπ ) = tg θ

sen ( nπ ) = 0
tg( nπ ) = 0

1

+ ( 2n − 1) = n 2

n

cos (θ + nπ ) = ( −1) cos θ

cos ( nπ ) = ( −1)

2
1. 5

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

( x1 + x2 +

tg (θ + π ) = tg θ
sen (θ + nπ ) = ( −1) sen θ

Gráfica 2. Las funciones trigonométricas csc x ,
s ec x , ctg x :

k =1

( a + b) ⋅ ( a − b) = a − b
2
( a + b ) ⋅ ( a + b) = ( a + b ) = a 2 + 2ab + b2
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 + 3ab2 + b3
3
( a − b ) = a3 − 3a 2b + 3ab2 − b3
2
( a + b + c ) = a 2 + b2 + c 2 + 2ab + 2ac + 2bc
2

tg ( −θ ) = − tg θ

tg (θ +2π ) = tg θ

s en x
co s x
tg x

sen α ⋅ cos β =

cos ( −θ ) = cosθ

cos (θ + 2π ) = cosθ

-6

5

tg α + tg β
ctg α + ctg β
FUNCIONES HIPERBÓLICAS

n

n

ALGUNOS PRODUCTOS
a ⋅ ( c + d ) = ac + ad

tg α ⋅ tg β =

sen (θ + π ) = − sen θ

-2
-8

n! = ∏ k

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

sen ( −θ ) = − sen θ

1 + ctg 2 θ = csc2 θ

sen (θ + 2π ) = sen θ

-1. 5

sen (α ± β )
cos α ⋅ cos β

tg 2 θ + 1 = sec2 θ

sen θ + cos2 θ = 1
2

-1

1
1
= 2 sen (α + β ) ⋅ cos (α − β )
2
2
1
1
= 2 sen (α − β ) ⋅ cos (α + β )
2
2
1
1
= 2 cos (α + β ) ⋅ cos (α − β )
2
2
1
1
= −2 sen (α + β ) ⋅ sen (α − β )
2
2

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 =...