Problemas resueltos de integral indefinida Matemáticas I Curso 2011-2012
35. Calcular las siguientes integrales
33
() ()
tt
t
por partes
tLnx xe
a Ln x dx t e dt
dx e dt
⎧⎫
⎪⎪
=→=
⎪⎪
===
⎨⎬
⎪⎪
=
⎪⎪
⎩⎭
∫∫
32
3
tt
u t du t dt
dv e dt v e
⎧⎫
⎪⎪
=→ =
⎪⎪
=
⎨⎬
⎪⎪
=→=
⎪⎪
⎩⎭
)
(
2
32 32
2
332
tt ttt
tt
por partes
ut du tdt
te tedt te te tedt
dv e dt v e
⎧⎫
⎪⎪
=→ =
⎪⎪
=− = =− − =
⎨⎬
⎪⎪
=→=
⎪⎪
⎩⎭
∫∫
(
)
32 32
36 36
tt t tt tt
tt
por partes
u t du dt
te te tedt te te te edt
dv e dt v e
⎧⎫
=→ =
⎪⎪
⎪⎪
−+ = =−+ − =
⎨⎬
⎪⎪
=→=
⎪⎪
⎩⎭
∫∫
()( )
32 32 3 2
366 366 3 6 6
tttt t
t e t e t e e C e t t t C x Ln x Ln x Ln x C−+−+=−+−+= − + −+
22
331
() 1
11
x
xx
x
te xLnt
ee tt t
bdx dt
ett
dx dt
t
⎧⎫
⎪⎪
=→=
⎪⎪
−−
⎪⎪
⎪⎪
===
⎨⎬
⎪⎪
++
=
⎪⎪
⎪⎪
⎪
⎭
⎪
⎩
∫∫
(1 3 ) 1
1
t
tt
−
+
∫
(1 3 )
1
t
dt dt
t
−
==
+
∫
(1 3 ) 4 1
334
31 4
111
3
11
Dr
Ddcr c
tdd
dt dt dt dt
t
ttt
tt
⎧⎫
⎪⎪
⎪⎪
=+→ =+
⎪⎪
⎛⎞
−⎪⎪
⎪⎪
⎟
⎜
== =−+=−+
⎟
⎨⎬
⎜⎟
⎜
⎪⎪
−+ ⎝ ⎠
+++
⎪⎪
=− +
⎪⎪
⎪⎪
++
⎪⎪
⎩⎭
∫∫∫∫
()
4
341 3 41 3 1
xx xx
tLntC e LneC eLne C=− + + + =− + + + = − + + +
()
1/2 6 1/2 5
4/3 3/2 6 4/3 6 3/2
65
3
.. .(2,3) 6 ()
() 6
() ()
6
mcm
xx t
cdxdx tdt
xx t t
xt dx tdt
xx x
⎧⎫
=⎪ ⎪
⎪⎪
== = =
⎨⎬
⎪⎪
++
=→ =
+⎪⎪
⎭
⎩
∫∫ ∫
38
56
89 89
1
66 6 61 61
1
tt
tdt dt dt Ln t C Ln x C
tt tt t
====++=++
+++
∫∫∫
2
()
() 2
Ln x Lnt
xt xt
ddx t
xdx t dt
⎧⎫
⎪⎪
=→ =
⎪⎪
==
⎨⎬
⎪⎪
=
⎪⎪
⎩⎭
∫
2t
∫
1
2
por partes
u Lnt du dt
dt Ln t dt t
dv dt v t
⎧⎫
⎪⎪
⎪⎪
=→=
⎪⎪
== =
⎨⎬
⎪⎪
⎪⎪
=→=
⎪⎪
⎩⎭
∫
1
22212222t Lnt t dt t Lnt dt t Lnt t C x Ln x x C
t
⎛⎞
⎟
⎜
=− =−=−+= −+
⎟
⎜⎟
⎜
⎝⎠
∫∫
