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F´ormula general (^2) ax+^ bx +^ c^ =^
0 ⇒^ x^
√−b±= (^2) b−^4 ac 2 a
Identidades Trigonom´
etricas
csc(θ) =^
(^1) sen(θ) sec(θ) =^
(^1) cos(θ) tan(θ) =^
sen(θ)cos(θ) cos( cot(θ) = θ)^1 =^ sen(θ)^ tan(θ
) (^2) sen(θ) + cos
2 (θ) = 1 (^2) tan(θ) + 1 = sec
2 (θ) (^2) cot(θ) + 1 = csc
2 (θ) sen(α^ ±^ β
) = sen(α
) cos(β)^ ±
sen(β) cos(
α)
cos(α^ ±^ β
) = cos(α) cos(
β)^ ∓^ sen(β
) sen(α)
tan(α^ ±^ β
tan(α) = )±tan(β) 1 ∓tan(α) tan(β) sen(2θ) = 2 sen(
θ) cos(θ) cos(2θ) = cos
2 (θ)^ −^ sen
2 (θ) = 1^ −
(^2) 2 sen(θ) =
cos(0)= 2 cos
2 (θ)^ −^1 tan(2θ) =
2 tan(θ) 21 −tan(θ) (^2) sen(θ) = 1 −cos(2θ) 2 (^2) cos(θ) = 1+cos(2θ) 2 sen(α) sen(
cos(αβ) = −β)−cos(α
+β) 2
cos(α) cos(
cos(αβ) = −β)+cos(α
+β) 2
sen(α) cos(
sen(αβ) = −β)+sen(α
+β) 2
cos(α)^ −^ cos(
β) =^ −2 sen
(^ )α+β^ sen^2
(^ )α−β^2
cos(α) + cos(
β) = 2 cos
(^ )α+β^ cos^2
(^ )α−β^2
sen(α) + sen(
β) = 2 sen
(^ )α+β^ cos^2
(^ )α−β^2
sen(α)^ −^ sen(
β) = 2 sen
(^ )α−β^ cos^2
(^ )α+β^2
α^ sen(
α)^ cos(
α)^ tan(
α)
◦^00
π◦ 306
√ 12 31 √ 2
3 π◦ 454 √^22
√^22
π◦ 603 √^32
π◦ 902
Propiedades de los Logaritmosc^ loga =^ c^ ⇔^ bb
=^ a log logc = (^) a cb (^) logab log(xy) = logb
(x) + logb
(y)b (^ )xlog^ = logby
(x)^ −^ logb
(y)b y^ log(x) =b y^ log(x)b √ylog( x) =b logxb^ y
Funciones hiperb´
olicas
senh(x) =
x−xe−e 2 cosh(x) =
x−xe+e 2 tanh(x) =
senh(x) cosh(x) (^2) cosh(x)^ −
(^2) senh(x) = 1 (^21) − tanh (θ) =^ sech
2 (θ) (^2) coth(x)^ −
(^2) 1 = csch (θ) Algunos L´ımites sen(x) l´ım^ x^ x→ 0
1 −cos(l´ım x→ 0 x)= 0^ x
1 −cos(l´ım x→ 0 x)^1 =^2 x^2 x e−^1 l´ım^ x^ x→ 0 = 1l´ım (1 + x→∞ )x^1 = l´ım x^ x→
(^1) x(1 + x) (^0) =^ e Tabla de Derivadas ′ım f (x) = l´^ h→
f^ (x+h)− 0 f^ (x)ım= l´ h h→
f^ (x)−f^ (hx ) x−h
D(uv) =^ uv
′^ ′+^ uv (^ )^ vuuD =^ v
′′−uv^2 v D(au) =^ au
′ n D(u) =^ nu n−^1 ′u u D(e) =^ e u′u u D(a) = ln(
u′a)au v^ D(u) =^ u v−^1 ′^ (uvln(
′u) + vu)
D(ln(u)) =
′u u D(log(u)) =a
′u u ln(a) D(sen(u)) = cos(
′u)u D(cos(u)) =
−^ sen(u)u
′ D(tan(u)) = sec
2 ′(u)u D(sec(u)) = sec(
u) tan(u)u
′ D(csc(u)) =
−^ csc(u) cot(
′u)u
D(cot(u)) =
(^2) − csc(u) ′u D(arc sen(
uu)) = ′ √^21 −u D(arc cos(
u)) =^ −^
′u √^21 −u D(arctan(
′uu)) = 1+u 2 D(arcsec(
uu)) = ′ √ (^2) uu−^1 D(senh(u)) = cosh(
′u)u D(cosh(u)) = senh(
′u)u D(tanh(u)) = sech
2 ′(u)u D(sech(u)) =
−sech(u) tanh(
′u)u
D(csch(u)) =
−csch(u) coth(
′u)u
D(coth(u)) =
(^2) −csch(u ′)u −^1 D(f (x)) =
(^1) ′−^1 f (f (x))
Tabla de Integrales ∫^ udv^ =^ uv
∫^ − vdu ∫^ nudu^ =^
n+1usi^ n^ n+^
6 =^ −^1
∫^1 du^ = ln^ u^
|u| ∫^ sen(u)du
=^ −^ cos(u
∫^ cos(u)du
= sen(u) ∫^ tan(u)du
= ln^ |^ sec(
u)|^ =^ −^ ln
|^ cos(u)|
∫^ sec(u)du
= ln^ |^ sec(
u) + tan(u
∫^ csc(u)du
= ln^ |^ csc(
u)^ −^ cot(u
∫^ cot(u)du
= ln^ |^ sen(
u)| ∫^2 sec(u)du
= tan(u) ∫^2 csc(u)du
=^ −^ cot(u
∫^ sec(u) tan(
u)du^ = sec(
u) ∫^ csc(u) cot(
u)du^ =^ −
csc(u) ∫^2 sen(u)du
sen(2u = − (^2) u) 4 ∫^2 cos(u)du
sen(2u = + (^2) u) 4 ∫^ uedu^ =^
ue ∫^ uadu^ =^
ua ln(a) ∫^ du=^22 a+u
( 1 arctan (^) a )ua ∫^ du=^22 a−u
∣∣ 1 u+ ln∣^2 a ∣∣a∣u−a ∫^ du=^22 u−a
∣∣ 1 u− ln∣^2 a ∣∣a∣u+a √∫ 2 u±^ a
√u 2 du = (^22) u±aa±^2
√ 2 ln |u+u 22 ±a| 2
√ ∫ 2 a−^ u
√u 2 du = 2 (^2 2) a−^ u+
(^2) a arc sen 2 (^ )u a
∫^ du√^22 a−u
= arc sen (^ )u a ∫^ du√^22 u±a
∣∣= lnu^ +∣ √^2 2 u±^ a
∫^ du^ √^2 uu−a
(^1) = arcsec 2 a^ (^ )u a ∫^ senh(u)du
= cosh(u
∫^ cosh(u)du
= senh(u
∫^2 sech(u)
du^ = tanh(
u) ∫^2 csch(u)
du^ =^ −^ coth(
u) ∫^ sech(u) tanh(
u)du^ =^ −^
sech(u) ∫^ csch(u) coth(
u)du^ =^ −^
csch(u) ∫^ mx(a^ +^
pnq^ bx) dx,^ (p, q) = 1 m+1Si ∈^ n^
q^ Z, t=^ a
n + bx m+1 Si +^ n^ pq^ ∈^ Z,^ tq^
−n^ = ax+ b
R(sen(x),^
cos(x)) (^ x t = tan 2 )^2 ;^ dx^ =^
dt^2 1+t sen(x) =^
2 t; cos(x^2 1+t
(^21) −t) = 2 1+t R(sen(x),^
cos(x)) =^
R(−^ sen(x
),^ −^ cos(x))
t^ = tan(x);
dt dx = 1+t 2 sen(x) =^
t√; cos(x^2 1+t
(^1) ) = √^2 1+t
Curvas Param´
etricas:^ r
(t) =^ x(t)i
+^ y(t)j
′^ ′ r=^ xi^ +
′ yj^ v^
′= r;^ T
′r = ;′ |r|^
a^ =^ r”;^
dym = =dx^ ′ydy ;^ ′^ x 2 ′m=^2 ′^ x dx
Volumen del s´
olido al rotar alrededor del eje: ∫^2 X: π f^ ( x)dx^ Y
∫^ : 2π xf^ ( x)dx
Longitud de arco
∫ √′^2 : x+
∫ √′ (^2) y ′^2 1 + y
´Area de la superficie rotar alrededor del eje:√∫^ X: 2π^ f^
′^2 1 + f Y
√∫ (^) : 2π x ′^2 1 + f √∫ (^) X: 2π y ′^2 ′^2 x+^ yY
√∫ (^) : 2π x ′^2 ′^2 x+^ y
Centroide∫^ M^ =^ f dx
∫^ ; M= (^) y xf dx;^ Mx
∫^12 = f^ dx 2
Myx = M^
Mxy = M Coordenadas Polares x^ =^ r^ cos(
θ);^ y^ =^ r
sen(θ);^ r
2 2 =^ x+^ y
2
^ arctan θ = (^ )^ y x
si^ x >^0 (^ yarctan x )^ +^ π^ si^
x <^0 πsgn(y) 2
si^ x^ = 0 Longitud:
∫^ √^2 r+^
′^2 ´rArea:
∫^12 rdθ^2
Pendiente:
r^ cos(θ)+r ′^ sen(θ)′ (^) rcos(θ)−r^ sen(θ) Coordenadas Esf´
ericas
x^ =^ p^ sen(
φ) cos(θ), y
=^ p^ sen(φ
) sen(θ)
z^ =^ p^ cos(
φ) Criterio de derivadas parciales Si^ ∇f^ (a, b
) =^ 〈^0 ,^0 〉,
2 ∂ A =
f(a, b)^2 ∂x (^2) ∂f B = ∂x∂y (a, b),^ C^
(^2) ∂f= (a, b^2 ∂y
),^ ∆ =^ B
2 −^ AC
∆^ >^0 ⇒^ (
a, b, f^ (a, b)) es un punto silla. ∆^ <^ 0 y^ A >
0 ⇒^ f^ (a, b
) es un m´
ınimo local.
∆^ <^ 0 y^ A <
0 ⇒^ f^ (a, b
) es un m´
aximo local. Series ∑nk^ =k=^
n(n+1) 2 ∑n^2 kk=^
n(n+1)(2= n+1) 6
∑n^3 kk=^
(^ n(n+1)=^2 )^2 ∑nk=
n+1k rr= (^) −^1 r− 1
∑∞k^ x^ k=^
(^1) = , si 1 −x^ |x|^ <^1
∑k∞x^ k=0^ k!^
x= e
∑(−1)∞^ k=
k^2 kx= cos( (2k)!^
x) ∑∞(−1)^ k=
k^2 k+1x= sen( (2k+1)!^
x) Ecuaci´on del Plano Pasa por (
x, y, z), perpendicular a (^000
A, B, C)
A(x^ −^ x) +^0
B(y^ −^ y) +^0
C(z^ −^ z) = 0^0 Producto Punto y Producto Cruz u · v =^ |u||v|^ cos(
α) proyA^ =B^
A·B B^2 |B|
compA^ =B^
A·B |B|
|A^ ×^ B|^ =
|A||B|^ sen(
α) (A^ ×^ B)^ ·^
^ a^1 C =
aa 2 3 bbb (^1 2 3) ccc 1 2 3
Ecuaci´on lineal de primer orden′^ y+^ py^ =^
−q; y= eh ∫^ ∫^ p;^ u^ =^
q;^ y^ =^ uyhyh^
+^ ky h
Ecuaci´on de Bernoulli
′^ y+^ py^ =
n qy. Sustituir
(^11) −n y = u .
Ecuaci´on exacta
M^ (x, y)dx
+^ N^ (x, y)
dy^ = 0 es exacta
⇔^ M=^ y^
N.x
Factor integrante para hacerla exacta:^ ∫^ μ(x) =^ e
My−Nx^ dx^ N^
o^ μ(y) =^ e
∫^ Nx−MydyM^
Ecuaci´on lineal con coeficientes constantes(n)^ αy+^ αn
(n−1)yn− 1 +^ · · ·^ +^ α^0
y^ = 0
Si^ r^ es ra´ız simple y real:
rx y = Ce
Si^ α^ ±^ βi^ son ra´
ıces simples
αx y = Ce 1 cos(βx) +
αx^ Cesen( 2 βx)
Si^ r^ es ra´ız de multiplicidad
m y^ =^ C^1
rx^ e+^ Cxe^2
rx^ +^ · · ·^ +
m−^1 Cxem rx
Ecuaci´on de Eulern(n)^ αxyn
n+ αxn− 1 −^1 (n−1)^ y+
· · ·^ +^ αn−
′^ xy+^ αy 10 = 0
Si^ r^ es ra´ız simple y real
r y = Cx
Si^ α^ ±^ βi^ son ra´
ıces simples:
α y = Cx 1 cos(β^ ln(x
α^ )) + Cx 2 sen(β^ ln(x))
Si^ r^ es ra´ız de multiplicidad
m:^ y^ =^ C
r^ x+^ Cx 12 r^ ln(x) +^ · · ·
r^ + Cxlnm m−^1
Formas de soluci´
on para^ α
(n)^ y+^ αnn
(n−1)^ y+− 1 · · ·^ +^ αy^0
=^ f^ (x)
Las ra´ıces del polinomio caracter´
ıstico son de multiplicidad
r
P^ ,^ Q,^ R^ y
S^ son polinomios de grado
m,^ n^ o^ k^
seg´un se indique.
k^ = m´ax^ {
m, n}
f^ (x)^
Ra´ıces^
Formas de la soluci´
on
Pn^
x^ = 0^
r^ xQn
αx Pe,^ α^ n real^
α^
r^ αxxQen
Pcos(βxm^
) +^ Qsen(n^
βx)^
±βi^
r^ x(Rcos(k^ βx) +^ Ssen(k^
βx))
αx e(Pcos(m^
βx) +^ Qsen(n^
βx))^ α^ +
r^ αx βi xe (Rcos(βxk^
) +^ Ssen(k^
βx))
Transformada de Laplace f (t)
L(f^ ) =^ F^ (
p)
f^ (t)^
∫^ ∞−ptef^ (^0
t)dt
f^ (t) +^ kg(t
)^
F^ (P^ ) +^ kG
(P^ )
1
1 ,^ p >^0 p^
αt^ e
1 ,^ p >^0 p−α^
sen(αt)^
α,^ p >^022 p+α
cos(αt)^
p,^ p >^022 p+α
senh(αt)^
α,^ p >^022 p−α
cosh(αt)^
p,^ p >^022 p−α
n t,^ n^ ∈^ N^
n!,^ p >^0 n+1^ p
r^ t,^ r >^0
Γ(r+1),^ p >r+1^ p
0
δ(t^ −^ α)^
−αpe,^ α > 0
H(t^ −^ α) =
{^1 t^ ≥^ α^0 t < α
−αp e,^ α >p^ 0
∫^ t f ∗ g = 0 f^ (x)g(t^ −^
x)dx^
F^ (p)G(p)
f^ (αt)^
(^ ) 1 pF ,^ α > α α 0
n tf^ (t)^
n(n)(−1)F^ (p),^ p >^0
(n) f (t)^
npF^ (p)^ −^ ∑nn−pk=^
k^ (k−1)f^ (0),
p >^0
f^ de periodo
T
∫^ T−ptf^ (t)e^0
dt−pT 1 −e
αt ef^ (t)^
F^ (p^ −^ α)
f^ (t^ −^ α)H
(t^ −^ α)^
−αpeF^ (p)
Funci´on Gamma ∫^ ∞ Γ(x) = 0 x−^1 −ttedt Γ(x^ + 1) =
xΓ(x) Si^ n^ ∈^ N^ , Γ(
n^ + 1) =^ n
!, Γ(0) = 1, Γ(
√/2) = π Variable Compleja z^ =^ x^ +^ yiz^ =^ x^ −^ yiRe(z) =^ x
,^ lm(z) =^ y
√; |z| = x 2 2 +^ y
arg(x^ +^ yi
) = 2πk^ +
(^ ^ arctan^
)^ ysix x >^0 (^ )yarctan x +^ π^ si^ x <
0 πsgn(y) 2
si^ x^ = 0 e^
y^6 = 0
n^ (rcis(θ)) n= rcis(nθ ) n^ w=^ rcis (θ)^ ⇒^ w^ =
√nθ+2 rcis(^
πk), k^ = 0n^
,^1 ,... ,^ (n^
−^ 1)
z^ x e=^ e(cos(
y) +^ i^ sen(
y)) log(z) = ln
|z|^ +^ iarg(
z) sen(z) = sen(
x) cosh(y) +
i^ cos(x) senh(
senh(izy) = ) i
cos(z) = cos(
x) cosh(y)^
−^ i^ sen(x) senh(
y) = cosh(
iz)
b^ b^ log( a=^ e a)^ Ecuaciones de Cauchy Riemman u=^ v,^ ux^ y^
=^ −vy x^ F´ormula de la integral de Cauchy ∫f^ (z)dz^ C^ z−z^0
= 2πif^ (z^0
),^ z∈^ Int^0
(C)
∫f^ (z)^ Cn(z−z)^0
2 πidz = (^) +1 n (n)f (z),^0! z∈^ Int(C^0
)
Residuo de f en un polo
zde orden^0
n
Res(f, z) =^0
1 l´ım(n−1)!^
n− dz→z 0 1 (f^ (z)(z^ −n−^1 dz
n z)) 0 Serie de Fourier f^ (x) definida en
[^ ]LL −^ ,^2 f^ (x)^ ≈^ a^0
∑^ (∞+ a^ n=
2 πcos( nxn L^
() + bsen (^) n ))^2 πnxL^
L ∫ (^12) a= 0 L − f^ (x)dx;L 2 L ∫ (^22) a= (^) n L − f^ (x) cos(L 2 2 π nx)L^ L ∫ (^22) b= (^) n L − f^ (x) senL 2 (^ )^2 π nxdxL^ Serie de Fourier en cosenos f^ (x) definida en [
, L] f^ (x)^ ≈^ a^0
∑^ (∞+ a^ n=
πcos( nxn L^ )) ∫^1 L a= 0 L^0 f^ (x)dx; ∫^2 L a= (^) n L^0
πf (x) cos( L nx) (^) Serie de Fourier en senos f^ (x) definida en [
, L] ∑∞ f (x) ≈ n
(bsen (^) n (^) = )πnxL^ ∫^2 L b= (^) n L^0
(^ πf (x) sen L )^ nxdx (^) Forma compleja de Fourier f^ (x) definida en [
−L, L], ∑inf ty f (x) = n
(^ iznπcen=−∞ x);L ∫^ L/ (^1) c= (^) n L 22 nπi/Lxf^ (x)e −L/^2
dx
Transformada
Z
xn^
Z{x}n xk
∞∑^ xkkz k= {^1 k^ =^ m^0 k^6 =^ m
1 m z C^
Cz,^ |z|^ >^ z−^1
1 k^ a
z,^ |z|^ >^ |z−a^
a| k^
z,^ |z|^ >^2 (z−1)
1 k−^1 ka
z,^ |z|^ >^2 (z−a)
|a|
cos(αk)^
z(z−cos(α)) (^2) z−^2 z^ cos(α)+
,^ |z|^ >^1 sen(αk)^
z^ sen(α) (^2) z−^2 z^ cos(α)+
,^ |z|^ >^1 xk+m^
mzZ{x} −k^
m−^1 ∑m xzn^ n=
−n
2 de marzo de 2021
