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FORMULARIO DE C^ ÁLCULO
D^ IFERENCIAL
E^ I^ NTEGRAL
VER^ .3.
Jesús Rubí Miranda (
jesusrubi1@yahoo.com
http://mx.geocities.com/estadisticapapers/ http://mx.geocities.com/dicalculus/^ VALOR ABSOLUTO
1
1 1 1
si^
si^
y 0 y^
n^ ó ó
n k^
k k^
kn nk
k k^
k
a^
a a^
a^ a a^
a a^ a
a^ a a^
a^
a
ab^
a b^
a^
a
a^ b
a^
b^
a^
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=^
= = = ≥ = −^
≥^
=^
⇔^
=^
+^ ≤
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∏^
(^ ) (^
p^ q^ )^ /
p^ q p
p^ q q qp^
pq p p^
p p^
p p q p^ q^
p a^ a
a a^
a a a^
a a b^
a^ b a^
a b^
b a^
^ ^
^ ^
EXPONENTES loglog^10
log^
log
log^
log^
log
log^
loglog^
ln
log^
log^
ln
log^
log^
y log
ln x a a^
a^
a
a^
a^
a
ra
a b a
b
e
N^ x
a
N
MN^
M^
N
M^
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N
NN^
r^
NN
N
N^
a^
a
N^
N^
N^
N
=^ ⇒
=^
= =^
=^
LOGARITMOS ALGUNOS PRODUCTOS^ (^
(^
) (^
(^
) (^
)^ (^
(^
) (^
)^ (^
(^
) (^
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(^
(^
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2 2
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3
2
2 3
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2 2
a^ c
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a^ b^
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a^ b^
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a^
ab^ b
x^ b^
x^ d^
x^
b^ d
x^
bd
ax^ b
cx
d^
acx^
ad^
bc x
bd
a^ b^
c^ d
ac
ad
bc
bd
a^ b^
a^
a b^
ab^
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a^ b^
a^
a b^
ab^
b
a^ b^
c^
a^
b^ c
⋅^ +
+^ ⋅
−^
=^
+^ ⋅
+^
=^ +
−^ ⋅
−^
=^ −
+^ ⋅
+^
=^ +
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+^ ⋅
+^
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+^ ⋅
+^
=^
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=^ +
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−^
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ab^
ac^
bc
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(^
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(^
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2
3 3
3
2
2 3
4 4
4 3
2 2
3 4
5
1 n^ n^ k^1
k^
n^ n
k a^ b^
a^
ab^
b^
a^ b
a^ b^
a^
a b^
ab^
b^
a^ b
a^ b^
a^
a b^
a b^
ab^
b^
a^
b
a^ b^
a^ b
a^
b^
n
−^ − = −^ ⋅
+^
=^
−^ ⋅
−^ ⋅
+^
+^
=^ −
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−^ ⋅
=^
−^
∀^ ∈
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∑^
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5
(^
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6
6
a^ b
a^
ab^
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a b^
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a^ b
a^
a b^
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ab^
b^
a^ b
+^ ⋅
+^ ⋅
+^ ⋅
−^
+^
=^ +
+^ ⋅
−^
+^
−^
=^
−^
CA
CO
HIP θ
Gráfica 4. Las funciones trigonométricas inversasarcctg
x^ , arcsec
x^ , arccsc
x^ :^
(^
)^
(^
(^
)^
(^
(^
)^
(^
(^
)^
(^
sen^
sen^
2sen
cos 2
sen^
sen^
2sen
cos 2
cos^
cos^
2 cos^
cos 2
cos^
cos^
2sen
sen 2
α^
β^
α^ β
α^ β
α^
β^
α^ β
α^ β
α^
β^
α^ β
α^ β
α^
β^
α^ β
α^ β
+^
=^
+^
⋅^
−^
=^
−^
⋅^
+^
=^
+^
⋅^
−^
= −^
+^
⋅^
-^
0
5
(^43210) -1 -
arc ctg xarc sec xarc csc x
(^
)^
(^ )
(^
)^
(^ )
1
1
1
1
1 (^1 )
impar
par
n^
k^ n
k^ k^
n^
n
kn^
k^ n
k^ k^
n^
n
k a^ b
a^ b
a^
b^
n
a^ b
a^ b
a^
b^
n
+^ −
−
+^ −
−
^ =
+^ ⋅
−^
=^
+^
∀^ ∈
^
^
^
+^ ⋅
−^
=^
−^
∀^ ∈
^
^
(^
(^
1 2
1
1 1
1 1
1
1 1
0
1
n n^
kk
n kn^
n k^
k
k^
k n^
n^
n
k^ k
k^
k
k^
k^
k
n k
k^
n
a^ a k
a^
a
c^ nc ca^
c^ a a^ b
a^
b
a^
a^
= a a
= =^
= =^
=^
= − +^ =
+^ +
= = +^
=^
−^
=^
∑ ∑^
∑^
∑^
SUMAS Y PRODUCTOS ∑
n
θ^
sen^
cos^
tg^
ctg^
sec^
csc
D^
∞^
1 2^
3 2^
D^60
3 2^
1 2^
D^90
∞^
(^
sen tg^
tg^
cos^
α^ β cos
α^
β^
α^ ± β
±^
=^
D 30 D 45
(^
)^
(^
(^
)^
(^
(^
)^
(^
sen^
cos^
sen^
sen
21
sen^
sen^
cos^
cos
21
cos^
cos^
cos^
cos
2 α^
β^
α^ β
α^ β
α^
β^
α^ β
α^ β
α^
β^
α^ β
α^ β
⋅^
=^
−^
+^
^
^
⋅^
=^
−^
−^
^
^
⋅^
=^
−^
+^
^
^
2
2 2
2 2
2 sen^
cos^
1 ctg
csc tg^ θ^1 sec
θ θ^
θ θ^
θ +^
+^
IDENTIDADES TRIGONOMÉTRICAS
[^ ]
[^ ]
sen^
cos^
tg^
ctg^
tg^
sec^
cos^
csc^
sen^
y^
x^ y y^
x^ y y^
x^ y y^
x^
yx
y^
x^
yx
y^
x^
π^ π π π^ π yx
π π π^ π ^
= ∠^
∈^ −^
^
= ∠^
= ∠^
∈^ −
= ∠^
= ∠^
= ∠^
= ∠^
∈ ^
= ∠^
= ∠^
∈^ −^
^
tg^
tg
tg^
tg^
ctg^ α^ β ctg α^
β^
α^
β
⋅^
=^
(^ ) sen (^ ) ( )
sen cos^
cos tg^
tg θ^
θ θ^
θ θ^
θ −^ = −−^ = − = −
senh^
cosh^
(^2) senh tgh^
cosh^1 ctgh^
tgh^1
sech^
cosh^1
csch^
x^ senh
x x^
x x^
x x^
x x^
x x^
x x x x x
e^ ex e^ ex x
e^
e
x^
x^ e
ee e x^
x^ e
e x^
x^ e
e x^
x^ e − − − − − −^ e
− − − =
=^
=^
=^
=^
=^
=^ +
=^
=^ −
FUNCIONES HIPERBÓLICAS
(^
(^
(^
(^
(^
(^
(^
)^ (^
(^
)^ (^
(^
sen^
sen
cos^
cos
tg^
tg sen^
sen
cos^
cos
tg^
tg sen^
1 sen
cos^
1 cos
tg^
tg
n n
θ^ π^ n n n
θ θ^
π^
θ
θ^
π^
θ θ^ π
θ
θ^ π
θ
θ^ π
θ θ^
π^
θ
θ^
π^
θ
θ^
π^
θ +^
+^
+^
+^
+^
+^
= + =^
+^
=^ −
+^
(^
)^
(^
(^
(^
(^
(^
(^
(^
1
1 1
2 1 2
3
2
1 3
4
3 2
1 4
5
4
3
1
2
1
k!
n
n^ k kn kn kn kn k n k
n
a^ k
d
a^ n
d n a^
l r^
a^ rl
ar^
a^
r^
r
k^
n^
n k^
n^
n^
n
k^
n^
n^ n k^
n^
n^
n^ n n^
n
n^
k n^
n = k
− = = = = =
+^ − =
=^
+^ −
^
^
^
^
^
^
−^
=^
=^
=^
+^
=^
+^
=^
+^
+^
+^ +^
+^ +
∑ ∑ ∑ ∑ ∑ ∑ =∏ ^ =^ ^
(^
)^
!^ ,!! n^0 n^
n^ k^ k k
k^ n n^ k^
k n x^ y
− x yk
+^
=^
^ ∑^
Gráfica 1. Las funciones trigonométricas:
cos^ x
, tg
x^ :
sen^ x
-8^ -
-^ -^
0 2
4
6 8
(^2) 1.5 (^1) 0.5 (^0) -0.5 -1 -1.5 -
sen xcos xtg x
[ { }^ ] { }^
senh :cosh :
tgh :^
ctgh :
,^1
sech :
csch :
→→^
→^ − −^
→^ −∞ −
∪^
→−^
→^
^
\
\ \ \ \ ^
\
(^ ) (^ )^
(^ )
(^ )
(^ )
sen^
cos^
tg^
sen^
cos^
tg^
n
n
n^ π n π n π n n n = =^ − = +π π π ^
=^ −
^
^
^
^
^
^
Gráfica 2. Las funciones trigonométricas
sec^ x^
, ctg
x^ :
csc^ x
Gráfica 5. Las funciones hiperbólicas
,^
senh^
x
cosh^
x^ tgh
x
(^
)^
1 2
1 2
1 2 1 2
k
n^
n n^ n
k^
k k n
x^ x
x^
x^ x
x
n^ n^
n
+^ +
+^
=^
"^
-8^ -
-^ -^
0 2
4
6 8
2.5^2 1.5^1 0.5^0 -0.5^ -1 -1.5^ -2 -2.
csc xsec xctg x
-^
0
5
(^543210) -1 -2 -3 -
senh xcosh xtgh x
sen^
cos^
cos^
sen^
π^2 θ^
θ π θ^
^ θ
=^
=^
(^
CONSTANTES^ 3.14159265359 π =2.71828182846 e =
(^
(^
)^ (^
(^
2
2 2 sen^222
sen^
cos^
cos^
sen
cos^
cos^
cos^
sen^
sen
tg^
tg
tg^
1 tg
tg
sen 2
2sen
cos cos 2
cos
sen 2 tg tg 2^
1 tg^1 sen^
1 cos 2 21 cos^
1 cos 2 21 cos 2 tg^ α^ β^1 cos 2
α^
β^
α^
β
α^ β
α^
β^
α^
β
α^
β
α^ β
α^ β θ^
θ^
θ
θ^
θ^
θ θ θ^
θ θ^
θ
θ^
θ θ
θ^
θ ±^
=^
±^
±^
=^ − =^
=^
=^ +
sen^
csc^
sen^1
cos^
sec^
cos
sen^
tg^
ctg
cos^
tg
COHIPCAHIP
COCA
θ^
θ^
θ
θ^
θ^
θ
θ θ^
θ
θ^
θ
=^
=^
=^
=^
TRIGONOMETRÍA
Gráfica 3. Las funciones trigonométricas inversasarcsen
x^ , arccos
x^ , arctg
x^ :
-^ -^ -^
0
1
2
3
(^43210) -1 -
arc sen xarc cos xarc tg x
radianes=180π
D
(^
(^
1
2
1
2
1 1
2
1
2
1 senh^
ln^
cosh^
ln^
1 ,^
tgh^
ln^
,^
ctgh^
ln^
,^
sech^
ln^
, 0^
csch^
ln^
,^
x^
x^
x^
x
x^
x^
x^
x x x^
x x x x^
x x
x
x^
x x x x^
x
x^
x
−= − − − − −
+^
+^
∀^ ∈
=^
±^
−^
=^
^
^
^
=^
^
^
^
±^
^
=^
<^ ≤
^
^
^
^
=^
+^
^
^
FUNCS HIPERBÓLICAS INVERSAS
\
(^ )^
(^ )^
(^ ) (
)^
(^ )(
(^ )^ (^
)(^
(^ )^
(^ )^
(^ )^
(^ )
(^ )^ (^ )
(^ )^
(^
0
0
0
0
0 0
0 2
2
3 3
5
7
(^2 ) ''
'^
: Taylor ! '' 0 0
' 0^
: Maclaurin !
1
2!^
3!^
sen^
3!^
5!^
7!^
cos
n
n n^
n n
x
n f^ x^ n
x^ x
f^ x^
f^ x^
f^ x^
x^ x f^ x
x^
x nf^
x
f^ x^
f^
f^
x f^
x n x^
x^
x
e^
x^
n
x^
x^
x^
x
x^ x
n
x
− −
=^
+^
−^
+^ + =
+^
+^ +
=^ +^
+^
+^
+^ +
=^ −
ALGUNAS SERIES
"^
"^ (
)^
(^
(^
)^
(^ ) ( )
2
4
6
2 2 1
2
3
4
1
3
5
7
(^2 )
2!^
4!^
6!^
ln 1^
tg^
nn nn nn 2 1
x^
x^
x^
xn
x^
x^
x^
x
x^
x^
n
x^
x^
x^
x
x^ x
−− − −− n
−^
+^
−^
+^ + −
+^ =
−^
+^
−^
+^ +
∠^
=^ −
(^
(^
sen^2 22
cos
sen
cos^
sen
cos
au
au
au
au
e^ a
bu
b^
bu
e^
bu du
a^ b e^ a
bu
b^
bu
e^
bu du
a^
− b
=^
=^
MAS INTEGRALES ∫ ∫
(^
(^
2 2
2
2
2 2 2
2
2
2
2 2
2
2
2
2
2
2
2
2
2
2
2
2
sen cos ln 1 ln 1 cos 1 sec
sen
2
2 ln 2
du^
ua
a^
u
ua
du^
u^
u^ a
u^
a du
u a u^ a^
u^
a^
a^
u
du^
a a^
u
u^ u^
a
u a^
a u^
a^
u
a^ u du
a^ u
a
u^
a
u^ a du
u^ a
u^
u^ a
= −∠=^
+^
±^
+^
=^ ∠
=^ ∠
−^
=^
−^
+^
±^
=^
±^
±^
+^
INTEGRALES CON ∫ ∫ ∫ ∫ ∫ ∫
(^
(^
2
2
2
2
2
2
2
2
2 2
1 tg 1 ctg^1 ln 21 ln 2 du^
u a^
a
u^
a
u a^
a
du^
u^ a
u
a a^ u
a
u^
adu
a^ u
u
a a^ a
u
a^
=^ u
=^
=^
INTREGRALES DE FRAC ∫ ∫ ∫−
(^
1
tgh^
ln cosh ctgh^
ln senh sech^
tg senh csch^
ctgh^
cosh 1 ln tgh
udu^
u udu^
u udu^
u
udu^
u − u = = = ∠= − =
2 (^ ) (^ ) ( )
2 2
2
cosh^2
senh^
1 tgh
sech ctgh^
1 csch senh^
senh cosh^
cosh tgh^
tgh x^
x x^
x x^
x x^
x x^
x x^
x −^
−^
−^ = −
IDENTIDADES DE FUNCS HIP
(^
(^
(^
2
2 2
senh^
senh^
cosh^
cosh^
senh
cosh^
cosh^
cosh^
senh^
senh
tgh^
tgh
tgh^
1 tgh
tgh
senh 2
2senh
cosh
cosh 2
cosh
senh 2 tgh tgh 2^
x^ y^^1 tgh
x^
y^
x^
y
x^ y
x^
y^
x^
y
x^
y
x^ y
x^
y
x^
x^
x
x^
x^
x x x^
x ±^
=^
±^
=^
±^
=^ ± = =
=^ +
(^
(^
(^222)
senh^
cosh 2
cosh^
cosh 2
(^2) cosh 2
tgh^
cosh 2
x^
x x^
x x
x^
x =^
=^
=^
senh 2 tgh^
cosh 2
x 1 x^
x =^
2
2
2
discriminante
ax^
bx^ c
b^
b^
ac
x^
a
b^
ac +^ +
= −^ ±
⇒^
OTRAS^ (^
lim 1^00001
lim 1senlim^
1 coslim
lim^
lim^
ln
x x
x x x x
x x x
x^
e e xx xx xe xx x
^
^
^
LÍMITES
(^ )^
(^
)^
(^ )
( ) (^ ) (^ ) (^
(^ )
0
0
lim^1
lim
x^
x^
x
n^
n
f^ x^
x^ f
x
df^
y
D f^
x^
dx^
x^
x
d^ cdxd^ cx
c dxd^ cx
ncx dxd^
du^
dv^
dw
u^ v
w dx^
dx^
dx^
dx
d^
du cu^
c dx^
∆ → dx
∆ →
−
+ ∆^
−^
=^
=^
∆^
= = = ±^ ±^
±^
=^
±^
±^
DERIVADAS
"^
(^ ) (^
)^ (^
)^ (
(^ )
2 1 n^
n d^
dv^
du
uv^
u^
v
dx^
dx^
dx
d^
dw^
dv^
du
uvw^
uv^
uw^
vw
dx^
dx^
dx^
dx
v du dx
u dv dx
d^ udx^ v
v
d^
du u^
nu dx^
− dx =^
=^
+^
^ =^ ^
2 1
2
(Regla de la Cadena) 1
donde
dF^
dF^ du dx^
du^
dx dudx^
dx dudF du dFdx^
dx du
x^ f
t
f^ t dy dt dydx^
dx dt
f^
t^
y^ f
t
=^
′^
=^
=^
′^
(^ ) (^
(^
(^ ) (^ ) (^ )^
1
ln
log log
log log^
0,^
ln a ln a u u u u v
v^
v
du dx d^
du
u dx^
u^
u^ dx
d^
e^ du u dx^
u^
dxe
d^
du u^
a^
a
dx^
u^
dx
d^
du e^
e dx^
dx d^
du a^
a^
a
dx^
dx
d^
du^
dv
u^
vu^
u u
dx^
dx^
dx
=^ −
=^ ⋅
=^
=^
⋅^
>^
=^ ⋅ =^
=^
+^
⋅^ ⋅
DERIVADA DE FUNCS LOG & EXP^ (^
(^
(^
(^
(^
(^
(^
2 sen^
cos cos^
sen tg^
sec ctg^
csc sec^
sec^
tg csc^
csc^
ctg
vers^
sen d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du u^
u^ u
dx^
dx
d^
du
u^
u^
u
dx^
dx
d^
du u^
u
dx^
dx = = − = = − = = − =
DERIVADA DE FUNCIONES TRIGO^ (^
(^
(^
(^
(^
(^
(^
sen^
cos
tg^
ctg^
si^
sec^
si^
si^
csc^
si^
vers^
d^
du
u dx^
dxu
d^
du
u dx^
dxu
d^
du u dx^
dxu
d^
du u dx^
dxu
u
d^
du
u^
u
dx^
dx u^ u
u
d^
du
u^
u
dx^
dx u^ u
d^
u dx^
u^ u
∠^
=^
∠^
= −^
∠^
=^
∠^
= −^
+^
∠^
= ±^
⋅^
−^
−^
∠^
=^
⋅^
+^
∠^
DERIV DE FUNCS TRIGO INVER
du ⋅ 2 dx
2 2 senh^
cosh cosh^
senh tgh^
sech ctgh^
csch sech^
sech^
tgh
csch^
csch^
ctgh
d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du u^
u
dx^
dx
d^
du
u^
u^
u
dx^
dx
d^
du
u^
u^
u
dx^
dx
= = = = − = − = − DERIVADA DE FUNCS HIPERBÓLICAS
1
2
1
2 1
2 1
2
1
1
senh^
si cosh
cosh^
,^
si cosh
tgh^
,^
ctgh^
,^
si sech
0,^
sech
d^
du u dx^
dxu
u
d^
du u^
u
dx^
dx^
u
u
d^
du u^
u
dx^
dxu
d^
du u^
u
dx^
dxu
u^
u
d^
du
u dx^
dx u^
u − − − −
−
+^
±^
=^
⋅^
>^
−^
−^
=^
⋅^
⋅^
−^
>^
=^
DERIVADA DE FUNCS HIP INV
1
1
2
si sech
0,^
csch^
,^
u^
u
d^
du
u^
u
dx^
dx u^
u
−
−
+^
<^
= −^
⋅^
(^
(^
)^
(^
(^
(^
ln
ln^
ln 1
ln^
ln^
ln^
log^
ln^
ln^
ln^
ln
log^
2 log^
ln^
2 ln^
u^
u u u
u u u^
u a a^
a
e du
e
aa a du^
aa a ua du
ua
a
ue du
e^
u udu^
u^ u
u^
u^
u
u
udu^
u^ u
u^
u
a^
a
u u^
udu^
u u u^ udu
u
=^
^ ≠ ^
=^
⋅^ −^
^
=^
=^
−^ =
=^
−^ =
=^
⋅^
=^
INTEGRALES DE FUNCS LOG & EXP ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫
(^ )^
(^ )
{^
}^
(^ )^
(^ )
(^ )^
(^ )
(^ )^
(^ )^
(^ )
(^ )^
(^ )
(^ ) (
)^
(^ )^
(^
(^ )^
[^ ]
(^ )^
(^ )
(^ )^
0 (^ )
,^ ,^
b^
b^
b
a^
a^
a
b^
b
a^
a
b^
c^
b
a^
a^
c
b^
a
a^
b a a
b a b^
b a^
a f^ x^
g^ x^
dx^
f^ x dx
g
x dx
cf^ x dx
c^
f^ x dx
c
f^ x dx
f^ x dx
f^ x dx
f^ x dx
f^ x dx
f^ x dxm b^ a
f^ x dx
M^
b^ a
m^
f^ x^
M^
x^ a b
m M
f^ x dx
g^ x dx f^ x^
g^ x^
x^ a ±^
=^
=^ ⋅^
=^
⋅^ −
≤^
≤^
⋅^ −
⇔^
≤^
≤^
⇔^
≤^
∀^ ∈
∫^
∫^
∫^
∫^
∫^
∫^
∫^
INTEGRALES DEFINIDAS,PROPIEDADES
\
\
[^ ]
(^ )^
(^ )^
si
b^
b a^
a
b
f^ x dx
f^ x^
dx^
a^ b
≤^
∫^
(^ )^
(^ )
(^
) (^
1
Integración por partes^1 1 nn ln adx^
ax af^ x dx
a^
f^ x dx u^ v
w^
dx^
udx^
vdx^
wdx
udv^
uv^
vduu u du
n n du^
u u
=
±^ ±
±^
=^
±^
±^
=^ − =^
∫ ∫^
∫^
∫^
∫^
INTEGRALES ∫ ∫
"^
sen^22
cos cos^
sen sec^
tg csc^
ctg sec^
tg^
sec csc^
ctg^
csc udu^
u udu^
u udu^
u udu^
u u^ udu
u u^
udu^
u = −= = = −=
INTEGRALES DE FUNCS TRIGO ∫ ∫ ∫ ∫ ∫ ∫^ tg^
ln cos
ln sec
ctg^
ln sen sec^
ln sec
tg
csc^
ln csc
ctg udu^
u^
u
udu^
u udu^
u^
u
udu^
u^
u
= −^
= =^
=^
(^
(^2222)
sen^
sen 2 2 41 cos^
sen 2 2 4 tg^
tg ctg^
u ctg udu^
u u udu^
u
udu^
u^ u udu^
u^ u =^ − =^ + = − = −^
∫ ∫ ∫ ∫^ sen
sen^
cos
cos^
cos^
sen
u^
udu^
u^ u
u
u^
udu^
u^ u
u =^
=^
∫ ∫ INTEGRALES DE FUNCS
(^
(^
2 2 2 2 2 2
sen^
sen^
cos^
cos^
tg^
tg^
ln^1
ctg^
ctg^
ln^1
sec^
sec^
ln^
sec^
cosh
csc^
csc^
ln^
udu^
u^
u^
u
udu^
u^
u^
u
udu^
u^
u^
u
udu^
u^
u^
u
udu^
u^
u^
u^
u
u^
u^
u
udu^
u^
u^
u^
u
∠^
=^ ∠
+^
∠^
=^ ∠
−^
∠^
=^ ∠
∠^
=^ ∠
+^
∠^
=^ ∠
−^
+^
=^ ∠
∠^
=^ ∠
+^
+^
TRIGO INV
csc^
cosh u^
u^
u
=^ ∠
INTEGRALES DE FUNCS HIP
senh^22
cosh cosh^
senh sech^
tgh csch^
ctgh sech^
tgh^
sech
csch^
ctgh^
csch
udu^
u udu^
u udu^
u udu^
u u^
udu^
u
u^
udu^
u
= = = = −
