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Integrales
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Apuntes acerca de las integrales
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TEMIA :CÁLCULO INTEGRAL DE UNA VARIABLE
FUNCION PRIMITIVA
Def : Sea f : ICIR -
IR una función real definida en un intervalo I .
Se dice que
La función Flx) es una función primitiva de la función flx) si
f- ( x ) f- ( x) tlx
'
=
,
c- I
Dado que dos funciones que se diferencian en una constante tienen la misma
derivada ,
si F es una primitiva de f en un intervalo I ,
entonces también lo
es G si es de la forma
G (x ) =
f- (x ) +
c
,
donde c es una constante arbitraria .
Def Sea f : : ICIR -
IR una función real definida en un intervalo I , se llama
función integral indefinida de f al conjunto de todas sus funciones primitivas
tal que
{ f- ( ) dx x = Flx) + c.
con c una constante arbitraria y
F una primitiva cualquiera de f.
TABLA DE INTEGRALES INMEDIATAS
/ Kdx kxtc { ¥ dx =
= lnxtc
{ Pdx ☒ {
"
P
✗ ✗
dx = + C =/ 1 COSX =
ser ✗ + C
p
-
{ e
✗
dx = e.
✗
+ c
fsenxdx = -
COSX + C
| +
DX
+ ×
,
= arctg ✗ + C
f
dx
1- ✗ a
= arcsenx + [
EJEMPLOS
• (( 5×2 + ✗ + 3) dx =
5¥ ¥ + +3×-1 C
'
/ COSCXIDX = Sen ( x ) + C
.
/ ( f- ✗
2
+
{ ✗ -
3) dx = 7- 1¥ +
12¥ - 3✗ =
¥ ¥ + -3×-1 a
•
} dx =
×
. COSX dx = In lsenxl +a
°
{ 2x Cas (✗
2
+ 5) dx = sen ( ✗ 2+5) +C
°
|,?# DX =
[ 2¥ ,
,
DX = arctg ( ✗ 2) + a
•
/ ¥2 dx =
arctg ✗ + c
•
/Y dx =
{ ✗ dx
, INTEGRACION POR '
ARTES
Sean U y o dos funciones derivable en un intervalo ] ,
se verifica que :
fulxldo = UCXIVCX) -
focx du )
Se utiliza en aquellos casos en los cuales podamos descomponer la función en dos
partes , y es consecuencia de la fórmula de la derivada de un producto .
Regla Alpes : Arcas Logarítmicas Polinómicas Exponenciales Senos y caseros
EJEMPLOS :
{
/✗ e
}
"
1)
le ex (
✗
✗ ex ,) + a
"
dx = u = ✗ → du = 1 ✗ l ' -
✗
DX =
-
e = × _
=
dv = e
✗ →
v = fexdx = ex
/
'
✗
{ }
'
2) ✗ e. dx = a = ✗
3
=/e "
¿%
"
dv = e. dx → o no es inmediata
{ }
2
U = ✗ → du = 2 ✗ DX
"
=/ édx
"
21/2
"
fe
dv = ✗e dx → o ✗e dx = ✗ =
,
12 )
"
/ fe
" " " "
#
2.
✗
fe 2 ✗ dx
¥ je
+ e a
e c
-
= = = +
.
- -
CAMBIO DE VARIABLES
Sea ¢ CN una función derivable en un intervalo [ , con derivada 104×1 continua
y sea
f una función continua ,
entonces haciendo el cambio de variable 1- =
Iocxl
se obtiene que
ff ( d ( xD .
#
'
( xtdx =
/ fctdt
Una vez calculada la integral respecto de la nueva variable , debemos deshacer
el cambio sustituyendo t por la inversa de la función ¢41 .
EJEMPLOS :
1) /☒ dx = / FÉ) dx = V9 IFÉI .
dx =3
/ FEF dx =
y Y
Sent cost + ser
'
✗ =: 1- sería cambio variable
Iz
1 casa de
pq
→ : →
= =
Es = Sent →
fgdx = castdt
DX =3
costdt
=3 /
Ttqrit .
3 castdt = 9 {cóítdt = a
/ 1+% dt g-(fldt -1%2 coscztdt) = =
T
'
Cos t
' + ser t = 1
cosa t -
= Cos (2t )
Zcaszt = 1 + casczt )
casa t
1+q 1- COSCZT )
=
→ serít =
2
G- (1- { sencsntl) G-Larsen(5) Isen( /¥))
=
+ + a + raras en + a
2)
{( e
"
-1×1+-3) dx =
fe "
dx +
/¥ ,
dx
T
=
/ 1 etdt,
+
¥ ,
dx =
jet + Inl ✗ +31 + C
3/1 = t
3d ✗ = dt
dztdesna-EI.ge
dx =
"
cambio
+ lnlxt 31 + C
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