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INTEGRAL DEFINIDA
( ) dxx
+2
1
31
( ) ( )
[ ]
4
81
4
46412684
12
31
4
12
2
232
4
2
2
3
4
133
2
3
4
2
1
2
3
4
2
1
23
=
++++=
++++=
+++=
+++=
I
I
I
xx
xx
I
dxxxxI
dxxx
0
2
243
[ ] ( )
[ ]
8
4
43
2
2
3
3
2
2
3
2
3
2
4
43
43
3
0
2
320
2
3
0
2
2
0
2
2
0
2
2
=
=
=
=
=
=
=
=
=
I
I
xuI
duuI
xdxdu
xu
xdxxI
dxxxI
dxxx
x
+64
1
31
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[ ] [ ] [ ]
( ) ( ) ( )
12
6215
2554
3
7215123
2
4
32
3
2
1
64
1
3 464
1
64
1
3
64
1
64
1
64
1
3/12/12/1
64
1
64
1
64
1
3
=
+=
+=
+=
+=
I
I
xxxI
dxxdxxdxxI
dxxdxx
dxxI
+
+4
2 6
1dx
x
x
[ ] [ ]
8
5ln55
5ln5110ln544ln525ln51
6ln56ln5
6
51
6
51
6
1
6
1
4
1
1
2
4
1
1
2
4
1
1
2
+=
+++++=
+++=
++
+=
+
++
+
+=
I
I
xxxxI
dxx
dxx
I
dxx
xdx
x
xI
+8
4
2 124 dxxx
3
368
3
56
3
256
3
56
1223
1223
1223
124124124
8
6
2
36
2
2
32
4
2
3
8
6
2
6
2
2
2
4
2
=
++=
+
=
++++
I
I
xxx
xxx
xxx
I
dxxdxxxdxxx i
dxxx
++4
4
26
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3
109
6
218
6
887159
3
44
6
71
6
59
623
623
623
666
4
2
232
3
233
4
23
4
2
2
2
3
2
3
4
2
=
=++
=++=
++
+
+=
+++++++=
I
I
xxx
xxx
xxx
I
dxxxdxxxdxxxI
( )dxxsenx 2/
0
cos
[ ] [ ]
( ) ( )
0
011
02
0cos2
cos
cos
cos
2/
0
2/
0
2/
0
2/
0
===
=
=
=
I
I
sensenI
senxxI
xdxsenxdxI
+4/
02
2
sec2
sec
x
xtgxdx
[ ]
32
1222
sec22
secsec2
4/
0
2
4/
0
2/1
4/
0
2
2
=
++=
+=
==
= +=
I
I
xI
duuu
duI
xtgxdxduxu
dxxxsen
2/
03cos5
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( ) ( )
+=
++=
2/
0
2/
0
2/
0
2/
0
22
18
2
1
352
135
2
1
xdxsenxdxsenI
dxxxsendxxxsenI
2
1
2
1
2
1
2
1
8
1
8
1
2
1
2
2cos
2
1
8
8cos
2
1 2/
0
2/
0
=
=
=
I
I
xxI
2
42
56xx
dx
3
14
3
2
4
=
+=
I
xarcsenI
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INTEGRALES IMPROPIAS
Determinar la convergencia o divergencia de las siguientes
integrales:
( )
+
+02/3
1x
dx
( )
2
01
12lim
12lim
lim
lim
1
1lim
0
0
2/3
0
2/3
0
2/3
=
=
=
=
=
=+=
+=
+
I
Iu
I
duuI
u
duI
dxdu
xu
x
dxI
t
t
t
t
t
t
t
t
t
Es convergente
1
101x
dx
100
100
1100lim
100lim
lim
lim
1
100
1
101
1
101
=
=
=
=
=
I
I
xI
dxxI
xdxI
t
t
t
t
t
t
Es convergente
dxex x+
3
2
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[ ] [ ]
0
03
1
3
1
11
3
111
3
1
lim,,
11
3
1lim
11
3
1lim
3
1lim
3
1lim
3
1lim
3
1lim
3
limlim
00
00
0
0
0
0
2
3
0
2
0
2 33
=
=+=
=
+
=
+=
+=
=
=
+=
+
+
+
+
+
I
I
eeeeI
iteelevaluando
eeeeI
eeI
duedueI
dxxdu
xu
dxexdxexI
tttt
tu
tt
u
t
t
u
t
t
u
t
t
x
tt
x
t
+
++ 222 xx
dx
( ) ( )
( )[ ] ( )[ ]
=
=+=
+++=
+++++=
+++
++=
+
+
+
I
I
xarctgxarctgI
x
dx
x
dxI
xx
dx
xx
dxI
t
tt
t
t
tt
t
t
tt
t
2
2
22
1lim1lim
11lim
11lim
22lim
22lim
0
0
0
2
0
2
0
2
0
2
+
+0
2
xx ee
dx
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[ ]
[ ]
2
4
2
2
2lim
12lim
12lim
12lim
2lim
0
0
0
2
0
2
0
0
=
=
=
=
+=
=
=
+=
+=
+=
+
+
+
+
+
I
I
arctgearctgeI
arctgeI
u
duI
dxedu
eu
e
dxeI
ee
dxI
ee
dxI
tx
t
t
t
x
x
t
x
x
t
t
x
xt
t
xxt
4
1
24x
dx
3
1ln
4
1
21
21ln
24
24ln
4
1lim
2
2ln
4
1lim
2lim
0
4
1
0
4
1
220
=
+
+
=
+
=
=
I
E
EI
x
xI
x
dxI
E
E
E
E
E
dxx
1
8
3/2
dxxI
E
E
=
1
8
3/2
0lim
[ ]
[ ]
( )
9
213
813lim
3lim
3lim
33
0
1
8
3
0
1
8
3/1
0
=+=
=
=
=
I
I
EI
xI
xI
E
E
E
E
E
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Es convergente
2
12
1xx
dx
[ ]
3
seclim
1
lim
2
10
2
1
20
=
=
=
I
xarcI
xx
dxI
E
E
E
E
+
1
2x
dx
1
1
11
1lim
lim
lim
1
1
2/1
1
2
=
=
=
=
=
+
+
+
I
tI
xI
dxxI
x
dxI
t
t
t
t
t
t
e
xxdx
1 ln
xdxdu
xu
xx
dxI
Ee
E
/
ln
lnlim
10
==
=
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[ ]
[ ]
2
ln2lim
2lim
lim
10
0
10
=
=
=
=
I
xI
uI
u
duI
Ee
E
Ee
E
Ee
E
Es convergente en 2
AREAS
!allar el "rea de la regi#n acotada $or 122 ++= xxy ,el e%e
& ' las rectas &()1, &(3
2
3
0
2
30
1
2
3
0
1
3
0
22
3
64
3
6421
3
1
23
1212
uA
A
xxx
xxx
A
dxxxxdxxxA
=
= +=
+++
++=
+++++=
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!alar el "rea de la regi#n * acotada $or3
2xY= el e%e ' ' las rectas &()1, &(1
2
1
0
40
1
4
0
1
1
0
33
1
12
1
2
1
22
22
uA
A
xxA
dxxdxxA
=
=+=
+
=
+=
!allar el "rea de la regi#n * acotada $or
32
3
433 xxxY =
, el e%e & ' las rectas &(0 , &(1
2
1
0
41
0
31
0
2
1
0
32
6
1
3
11
2
3
43
4
3
3
2
3
3
433
uA
A
xxxA
dxxxxA
=
=
=
=
!allar el "rea de la regi#n * limitada $or2xxY = el e%e
&
2
1
0
32
1
0
2
6
1
3
1
2
1
32
uA
A
xxA
dxxxA
=
=
=
=
!allar el "rea de la +igura limitada $or la curva xy =3
la recta '(1 , la vertical &(8
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[ ]2
8
1
3 4
8
1
3/1
4
17uA
xA
dxxA
=
=
=
!allar el "rea de la +igura limitada $or la curva ( )( )21 =
xxxy ' el e%e &
2
2
1
23
41
0
23
4
2
1
23
1
0
23
2
1
4
1
4
1
44
2323
uA
A
xx
x
xx
x
A
xdxxxdxxxxA
=
+=
+
+=
++=
Encontrar el "rea de la regi#n acotada $or la curva 3
2
=x
yel e%e & ' las rectas &(4 ' &( 5
( )[ ]
( ) ( )2
5
4
5
4
5
4
2ln2
34ln235ln2
3ln2
32
3
2
uA
A
xA
x
dxA
x
dxA
=
=
=
=
=
alcular el "rea de la +igura com$rendida entre la curva tgxy= ,
el e%e & ' la recta 3
=x
( )
2
3/
0
3/
0
2ln
0sec3
secln
secln
uA
A
xA
tgxdxA
=
=
=
=
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!allar el "rea de la regi#n limitada $or las gra+icas de32
, xyxy == , las rectas &()1 ' &(2
2
2
1
341
1
43
1
1
2
1
2332
2
25
2
25
2
17
12
8
3443
uA
A
xxxx
A
dxxxdxxxA
=
=+=
+
=
+=
!allar el "rea de la regi#n * limitada $or las gr"+icas de2
4 xy = ' ( )32ln = xy
( )
( )
2
1
0
2/3
1
0
3
1332
2
273
23
2
43
23
2
42
3
2
3
32ln
ue
A
eA
yye
A
dyye
A
xe
xy
y
y
y
+=
+=
+=
+
=
=+
=
LONGITUD DE ARCO
ule la longitud del arco de la $ar"-ola semic.-ica( )32 1
3
2= xy
com$rendida dentro de la $ar"-ola 3
2 xy =
( )
( ) ( )
( )
+=
=
+=
2
1
2
2/1/
3
12
3
3
21
12
3
3
2
13
2
dxxl
dxxxf
xfx
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( ) ( )
( ) ( ) ( )
[ ]
[ ]ul
l
xl
dxx
l
dxx
dxxl
225529
4
2225529
2
253
2
23
113
3
2
23
1
2
13
2
1
12
11811
4
9
3
21
2/32/3
2
1
2/3
2
1
2
1
2
1
=
=
=
=
=
+=+=
alcular la longitud de la curva ( )22
39 axxay = desde &(0 asta &(3
( )
( )
( )
aul
xax
a
l
dxx
ax
adx
ax
axl
dxax
aaxxdx
ax
axl
dxax
axy
a
axxa
axaxy
a
aa
aa
34
32
1
2
1
41
4
21
21
2
9
318
1293
3
0
2/3
3
0
3
0
2
3
0
223
0
2
/
2
22
/
=
+=
=
+=
++=
+=
=
+=
!allar la longitud de la curva22
4 xxy = com$rendida entre los dos $untos en ue se corta al e%e
&
( )
=
++
=
++=
=
2
02
2
0
2
22
2
0
2
2
2
/
44
4
444
4
441
4
2
xx
dxl
dxxx
xxxxl
dxxxxxl
dxxx
xy
ul
l
xarcsenl
2
02
4
2
4
2
0
=
=
=
