Formulario

Jannet Haidee Ortiz Madin

January 15, 2007

Identi Trigono
sin θ = cat.op
hipo
cos θ = cat.ad
hipo
cat.op
tan θ = cat.ad
1
csc θ = sin
Funciones Hiperbolicas
θ
1
sec θ = cos θ x −x
sin θ
tan θ = cos sinh θ = e −e
2
θ
x −x
sin2 θ + cos2 θ = 1 cosh θ = e +e
2
sec2 θ − tan2 θ = 1 x −x
csc2 θ − cot2 θ = 1 tanh θ = ex −e−x
e +e
sin(−θ) = − sin θ x −x
cos(−θ) = cos θ coth θ = ex +e−x
e −e
tan(−θ) = − tan θ sechθ = 2
sin(θ + 2π) = sin θ ex +e−x
cos(θ + 2π) = cos θ cschθ = x 2 −x
e −e
tan(θ + 2π) = tan θ
cosh2 x − sinh2 x = 1
sin(θ + π) = − sin θ
tanh x + sech2 x = 1
2
cos(θ + π) = − cos θ 2 2
coth x − csch x = 1
tan(θ + π) = tan θ
sinh(−θ) = − sinh θ
sin(θ + nπ) = (−1)n sin θ
cosh(−θ) = cosh θ
cos(θ + nπ) = (−1)n cos θ
tanh(−θ) = − tanh θ
tan(θ + nπ) = tan θ
sinh(α ± β) = sinh α cosh β ± cosh α sinh β
sin(α ± β) = sin α cos β ± cos α sin β
cosh(α ± β) = cosh α cosh β ± sinh α sinh β
cos(α ± β) = cos α cos β ∓ sin α sin β tanh α±tanh β
tan α±tan β tanh(α ± β) = 1±tanh
tan(α ± β) = 1∓tan α tan β
α tanh β
sinh 2θ = 2 sinh θ cosh θ
sin 2θ = 2 sin θ cos θ
cosh 2θ = cosh2 θ + sinh2 θ
cos 2θ = cos2 θ − sin2 θ
tanh 2θ = 2 tanh2θ
cos 2θ = 1 − 2 sin2 θ 1+tanh θ
cos 2θ = 2 cos2 θ − 1 sinh2 θ = 1 (cosh 2θ − 1)
2
tan 2θ = 2 tan2θ
q1−tan θ cosh2 θ = 2
1 (cosh 2θ + 1)

sin θ
2
= 1−cos θ
2
tanh2 θ = cosh 2θ−1
cosh 2θ+1
q
θ 1+cos θ
cos 2 = 2
tan θ
2
= 1−cos
sin θ
θ

sin2 θ = 1
2
(1 − cos 2θ)
cos2 θ = 1 (1 + cos 2θ)
2
tan2 θ = 1−cos 2θ
1+cos 2θ Funciones Hiper Inv
sin α + sin β = 2 sin α+β
2
cos α−β
2
sin α − sin β = 2 cos α+β sin α−β sinh−1 θ = ln(θ + θ 2 + 1)
p
2 2
cos α + cos β = 2 cos α+β cos α−β cosh−1 θ = ln(θ + θ 2 − 1)
p
2 2
cos α − cos β = 2 sin α+β
2
sin α−β
2 tanh −1 1 1+θ
θ = 2 ln( 1−θ )
sin(α±β)
tan α ± tan β = cos α cos β coth−1 θ = 2
1 ln( θ+1 )
θ−1
sin α sin β = 1
q
2
[cos(α − β) − cos(α + β)] 1± 1−θ 2
sech−1 θ = ln( )
cos α cos β = 1
2
[cos(α + β) + cos(α − β)] θq
1 [sin(α + β) + sin(α − β)]
sin α cos β = 2 θ 2 +1
csch−1 θ = ln( θ
1 +
θ
)
1 [sin(α + β) − sin(α − β)]
cos α sin β = 2

1
Logaritmos Vectores
q
loga N = x −→ ax = N modulo −→ ā = a2 2 2
1 + a2 + a3
loga M N = loga M + loga N
prodesc −→ ā.b̄ = a1 b1 + a2 b2 + a3 b3
loga M
N
= loga M − loga N prodesc −→ ā.b̄ = āb̄ cos θ; 00 ≤ θ ≤ 1800
loga N r = r loga N compvect −→ ā = ā. b̄ b̄
log N b b
loga N = logb a compesc −→ ā = ā. b̄
b b
ā.b̄
∠ ÷ vect −→ θ = ∠ cos a.b
a
cos directores −→ α = ∠ cos a1
a
β = ∠ cos a2

Sumas y Productos γ = ∠ cos a3

a

cos2 α + cos2 β + cos2 ˛γ = 1 ˛

˛ î ĵ k̂ ˛˛
a1 + a2 + · · · + an = Σak ˛
Σc = nc prodvect −→ ā × b̄ = ˛˛a1 a2 a3 ˛˛
˛ b1 b2 b3 ˛
Σcak = cΣak
Σ(ak + bk ) = Σak + Σbk ā × b̄ = āb̄ sin θ; 00 ≤ θ ≤ 1800
n(n+1)
Σi = 2
2 n(n+1)(2n+1)
Σi = 6
n(n+1) 2
Σi3 = [ 2
]
5 4 3
Σi4 = 6n +15n30+10n −n
1 + 3 + 5 + · · · + (2n + 1) = n2
Calculo
T eormV alorM edCalcIntrg
Rb
f (c) = a f (x)dx
(b−a)
T eormF undCalcIntrg

Geometria Analitica
Rb
a f (x)dx = F (b) − F (a)
y = uv −→ y , = uv , + u, v
, ,
a b c y= u v
−→ y , = vu −uv
2
ley. sin −→ sin α
= sin β
= sin γ
v

ley. cos −→ a2 = b2 + c2 − 2bc cos α

ley. cos −→ b2 = a2 + c2 − 2ac cos β
ley. cos −→ c2 = a2 + b2 − 2ab cos γ
recta −→ y = mx + b
m = pdte
b = ord
y −y
Deriv Funcs Trigo
pdterecta −→ m = x1 −x2
1 2
ecgralAx + By + C = 0 d sin u = cos u du
Ax1 +By1 +C dx dx
distd = q d cos u = − sin u du
A2 +B 2 dx dx
2
d tan u = sec2 u du
circunf −→ x + =r y2 2 dx dx
C(h, k) −→ (x − h) + (y − k)2 = r 2
2 d
dx
cot u = − csc2 u du
dx
d sec u = sec u tan u du
ecgralx2 + y 2 + Dx + Ey + F = 0 dx dx
parabx −→ y 2 = 4px d csc u = − csc u cot u du
dx dx
paraby −→ x2 = 4py
C(h, k)enx −→ (y − k)2 = 4p(x − h)
C(h, k)eny −→ (x − h)2 = 4p(y − k)
2 2
elipx −→ x2 + y2 = 1
a b

b
2
a
2
elipy −→ x2 + y 2 = 1
(x−h)2 (y−k)2
Deriv Fun Trig Inv
C(h, k)enx −→ + =1
a2 b2
d ∠ sin u = q 1 du
(x−h)2 (y−k)2 dx dx
C(h, k)eny −→ + =1 1−u2
b2 a2
2
hiperx −→ x2 − y = 1
d ∠ cos u = −q 1 du
dx dx
a2 b2 1−u2
y 2 2
hipery −→ − x2 = 1 d ∠ tan u = 1 du
a2 b dx 1+u2 dx
(x−h)2 (y−k)2 d ∠ cot u = − 1 du
C(h, k)enx −→ − =1 dx 1+u2 dx
a2 b2
(y−k)2 (x−h)2 d ∠ sec u = q1 du
C(h, k)eny −→ − =1 dx
u u2 −1
dx
a2 b2
d ∠ csc u = − q 1 du
dx dx
u u2 −1

2
Deriv Fun Log y Exp Integr Fun Trigo
d ln u = 1 du R
dx u dx sin udu = − cos u + c
d log u = log e du R
dx u dx R cos udu = sin u + c
d log u = loga e du tan udu = − ln cos u + c = ln sec u + c
dx a u dx
R
d eu = eu du R cot udu = ln sin u + c
dx dx sec udu = ln sec u + tan u + c
d au = au ln a du
R
dx dx
csc udu = ln csc u − cotu + c
sec2 udu = tan u + c
R
d uv = uv−1 vu, + uv ln uv ,
dx
csc2 udu = − cot u + c
R
R
R sec u tan udu = sec u + c
csc u cot udu = − csc u + c
sin2 udu = u − 1
R
2 4
sin 2u + c
cos2 udu = u
Deriv Funcs Hiperb + 1
R
2 4
sin 2u + c
q du = ∠ sin u
R
a
+c
a2 −u2
d sinh u = cosh u du
R du 1 u
= a ∠ tan a + c
dx dx u2 +a2
d cosh u = sinh u du
q du 1 ∠ sec u +
R
dx dx = a a
c
d tanh u = sech2 u du u u2 −a2
dx dx R du 1 ln u−a
d coth u = −csch2 u du = 2a +c
dx dx u2 −a2 u+a
d sechu = −sechu tanh u du
R du 1 ln a+u
= +c
dx dx a2 −u2 2a a−u
d cschu = −cschu coth u du p
q du u2 + a2 + c
R
dx dx = ln u +
u2 +a2
p
q du u2 − a2 + c
R
= ln u +
u2 −a2
sin(nx) cos(mx)dx = 1
R R
2
[sin(nx + mx) + sin(nx − mx)]dx

Der Fun Hiper Inv 1 R [cos(nx − mx) − cos(nx + mx)]dx

R
sin(nx) sin(mx)dx = 2
cos(nx) cos(mx)dx = 1
R R
2
[cos(nx − mx) + cos(nx + mx)]dx

d
dx
sinh−1 u = q 1 du
dx
u2 +1
d
dx
cosh−1 u = q 1 du ; u
dx
>1
u2 −1
d tanh−1 u = 1 du ; u < 1
dx 1−u2 dx
d coth−1 u =
dx
1 du ; u > 1
1−u2 dx
d sech−1 u = − q 1 du ; 0 <
Int Funcs Hip
dx dx
u<1
u 1−u2
R
d csch−1 u = − q 1 du ; u 6= 0 sinh udu = cosh u + c
dx dx R
u 1+u2 R cosh udu = sinh u + c
R tanh udu = ln cosh u + c
coth udu = ln sinh u + c
sechudu = tan−1 sinh u + c
R
cschudu = ln tanh u
R
2
+c

Integrales R
R
R
sech2 udu = tanh u + c
csch2 udu = − coth u + c
Rb sechu tanh udu = −sechu + c
cf (x)dx = c ab f (x)dx
R R
cschu coth udu = −cschu + c
Rab
f (x)dx = a f (x)dx + cb f (x)dx q du = sinh−1 u
Rc R R
a
+c
Rab Ra u2 +a2
f (x)dx = − f (x)dx
Raa b
p
q du u2 + a2 + c
R
f (x) = 0 = ln u +
Ra u2 +a2
adx = ax + c
q du = cosh−1 u
R
R R +c
R af (x)dx = a f (x)dx R R R u2 −a2
a
R (u ± v ± w)dx R = udx ± vdx ± wdx q du
p
= ln u + u2 − a2 +
R
udv = uv − vdu; (porpartes) c
R n n+1 u2 −a2
u du un+1 + c; n 6= −1 R du 1 ln a+u + c
= 2a
R du a2 −u2 a−u
= ln u + c du 1 csch−1 u + c
R u
R
= −a
eu du = eu + c a
q
u a2 +u2
R u au + c; a > 0, a 6= 1 1 sinh−1 a + c
a du = ln q du
R
a = −a u
R u
ua du = ln au (u − 1 ) + c u a2 +u2
a ln a 1 sech−1 u + c
q du
R
R u u = −a
R ue du = e (u − 1) + c u a2 −u2
a
R ln udu = u(lnuu − 1) + c 1 csch−1 a + c
q du
R
loga udu = ln a (ln u − 1) + c = −a u
2 u a2 −u2
u loga udu = u4 (2 loga u − 1) + c
R
R u 2
u ln udu = 4 (2 ln u − 1) + c

3
Solid, Volmn
V ol.Sold.Rev.
Eje.rev.x −→ V = π ab [f (x)]2 dx
R

Eje.rev.y −→ V = π b [f (y)]2 dy
Ra

Deriv Parc Mixtas

δ ( δ ) δ2 f
δy δx
= δyδx = fyx (x, y)
δ ( δ ) δ2 f
δx δy
= δxδy = fxy (x, y)
δ f2 2
δ f
Si∃Der.P ar −→ δyδx = δxδy
Derv.P ar.Suces
δ [· · · δ ( δf (x,y) ) · · · ] = δ n z
δx δx δx δxn
δ [· · · δ ( δf (x,y) ) · · · ] = δ n z
δy δy δy δy n
Der.P ar.M ixt.Suces
2 δ 3 f (x,y)
δ [ δ f (x,y) ] = δxδxδy = fxxy (x, y)
δx δxδy
2 δ 3 f (x,y)
δ [ δ f (x,y) ] = δyδxδy = fyxy (x, y)
δy δxδy
CE −CA
Dirf.T otal −→ Erel = CA
× 100
δz
Dif ern −→ dz = δx dx + δyδz dy

Reg.Caden −→ dxdz = δz dr + δz ds + · · · + δz dw
δr dx δs dx δw dx
dy F (x,y)
Der.Implic −→ dx = − Fx (x,y)
y
δz = − Fx (x,y,z)
Der.Implic −→ δx Fz (x,y,z)
δz = − Fy (x,y,z)
Der.Implic −→ δy Fz (x,y,z)
Derv.Direcc −→ dz
ds
= δxδz cos θ + δz sin θ
δy
Der.Dir. −→ Du f (x, y) = fx (x, y)u1 + fy (x, y)u2