CalA 6 4 Respostas

Prévia do material em texto

451 
6.4 – EXERCÍCIOS – pg. 250 
 
 Calcular as integrais seguintes usando o método da substituição. 
 
1. ∫ +−+ dxxxx )12()322( 102 
.
11
)322(
2
1)12()322(
:
)12(2)24(
322
:
112
102
2
c
xxdxxxx
Temos
dxxdxxdu
xxu
seFazendo
+
−+
=+−+
+=+=
−+=
−
∫
 
2. ∫ − dxxx
27/13 )2( 
( ) ( ) .2
24
7
7
8
2
3
1)2(
:
3
2
:
7
87
8
3
3
27/13
2
3
cxc
xdxxx
Temos
dxxdu
xu
seFazendo
+−=+
−
=−
=
−=
−
∫
 
3. ∫
−
5 2 1x
dxx
 
( )
cxc
x
x
dxx
Temos
dxxdu
xu
seFazendo
dxxx
+−=+
−
=
−
=
−=
−
−
∫
∫
−
5
4
5
1
)1(
8
5
5
4
1
2
1
1
:
2
1
:
)1(
2
5/42
5 2
2
2
 
4. ∫ − dxxx
2345 
mathe
Realce
mathe
Realce
 452 
( )
( ) ( ) .34
9
5
2
3
34
6
1
.5345
:
6
34
:
345)34(5
2
32
3
2
1
2
1
2
2
2
2
22
cxc
xdxxx
Temos
dxxdu
xu
seFazendo
dxxxdxxx
+−
−
=+
−−
=−
−=
−=
−
−=−=
∫
∫∫
 
5. ∫ + dxxx
42 2 
( )
( )
( ) cx
c
x
dxxx
++=
+
+
=
+= ∫
2
3
2
3
2
1
2
2
2
21
6
1
2
3
21
4
1
21
 Fazendo: 
dxxdu
xu
4
21 2
=
+=
 
6. ∫ + dtee
tt 22 31)2( 
( ) ( ) .2
8
3
3
4
2
2
1)2(
:
2
2
:
3
4
3
1 23
4
2
22
2
2
cec
edtee
Temos
dtedu
eu
seFazendo
t
t
tt
t
t
++=+
+
=+
=
+=
−
∫
 
7. ∫ + 4t
t
e
dte
 
. e 4 que sendo , 4ln dtedueuce
u
du ttt
=+=++== ∫ 
8. ∫
+ dx
x
e x
2
/1 2
 
mathe
Realce
mathe
Realce
 453 
.
1
.
:se-doConsideran
.
2
1
.221
2
1
2
2
1
1
111
x
edu
eu
c
x
ec
x
edxxdx
x
e
x
x
xxx
−
=
=
+−−=+
−
+−=+=
−
−
∫∫
 
9. ∫ dxxxtg
2sec 
c
xtg
+=
2
2
. considerando-se: 
dxxdu
xtgu
2sec=
=
 
10. ∫ dxxxsen cos
4
 
c
xsen
+=
5
5
 considerando-se: dxxdu
xsenu
cos=
=
 
11. ∫ dxx
xsen
5cos
 
cx
c
x
x
dxxsenx
+=
+=
−
−=
=
−
−
∫
4
4
4
5
sec
4
1
cos4
1
4
cos
.cos
 utilizando: 
senxdxdu
xu
−=
= cos
 
12. ∫
− dx
x
xxsen
cos
cos52
 
cxx
dx
x
xsen
+−−=
−= ∫∫
5|cos|ln2
5
cos
2
 utilizando: 
senxdxdu
xu
−=
= cos
 
13. ∫ dxee
xx 2cos 
.2
2
:se-doConsideran
.2
2
1
dxedu
eu
cesen
x
x
x
=
=
+=
 
mathe
Realce
mathe
Realce
 454 
14. ∫ dxx
x 2cos
2
 
.2
:se-doConsideran
4
1
2
1
2
1
2
22
dxxdu
xu
cxsencxsen
=
=
+=+=
 
15. ∫ − θpiθ dsen )5( 
( )
.5
5
:se-doConsideran
.5cos
5
1
θ
piθ
piθ
ddu
u
c
=
−=
+−−=
 
16. dy
y
ysenarc
∫
−
212
 
( ) ( )
.
1
1
:se-doConsideran
.
4
1
22
1
2
2
2
dy
y
du
ysenarcu
cysenarccysenarc
−
=
=
+=+=
 
17. ∫ + θθ
θ d
tgba
2sec2
 
Ctgba
b
++= ||ln1.2 θ 
Considerando-se: 
θθ
θ
dbdu
tgbau
2sec.=
+=
 
18. ∫ + 216 x
dx
 
mathe
Realce
mathe
Realce
 455 
c
x
tgarccxtgarc
x
dx
+=+=






+
= ∫ 44
1
4
4
16
1
4
1
16
1
2 , utilizando: 
dxdu
x
u
4
1
4
=
=
 
19. ∫ +− 442 yy
dy
 
c
y
c
ydyy
y
dy
+
−
=+
−
−
=−=
−
=
−
−
∫∫ 2
1
1
)2()2()2(
1
2
2 , utilizando: dydu
yu
=
−= 2
 
20. ∫ θθθ dsen cos3 
( ) .
4
3
3
4
)(
cos 3
4
3/4
3/1
csenc
sendsen +=+== ∫ θ
θθθθ 
21. ∫ dxx
x2ln
 
( ) ( ) ( )
.
22
ln
:se-doConsideran
.lnln4
4
1)(ln
4
1
2
ln
2
1
2
2
2222
22
dx
xx
xdu
xu
cxcxcxc
x
==
=
+=+=+=+
 
 
22. dxee axax 2)( −+∫ 
( )
( ) .222
2
1
2
12
2
12
22
22222
cx
a
axhsen
cxee
a
ce
a
xe
a
dxee
axax
axaxaxax
++=++−=
+−+=++=
−
−−
∫
 
23. ∫ + dttt
243 
mathe
Realce
mathe
Realce
 456 
( ) ( ) ( )
( ) ( ) .13.
9
113.
2
3
.
6
1
2
3
13
6
11313
2
3
2
3
2
3
2
1
22
2
222
ctct
c
tdtttdttt
++=++=
+
+
=+=+= ∫∫
 
Considerando-se: 
dttdu
tu
6
13 2
=
+=
. 
24. ∫ ++ 34204
4
2 xx
dx
 
.
3
2
52
3
2
2
3
2
5
2
3
1
2
3
2
5 22
c
x
tgarcc
x
tgarc
x
dx
+






+
=+






+
=






+





+
= ∫ 
25. ∫ +− 14
3
2 xx
dx
 
( ) ( ) ( )∫∫∫ −
−
−=
−
−
−=
−−
=
3
213
13
3
2
3
3
33
32
3
222
x
dx
x
dx
x
dx
 
.
23
23ln
2
3
3
21
3
21
ln
2
13 c
x
x
c
x
x
+
−+
−+−
=+
−
−
−
+
−= 
Considerando-se: 
( )
dxdu
x
u
x
u
3
1
3
2
3
2 22
=
−
=
−
=
 
Resposta alternativa: 
mathe
Realce
 457 
.1
3
2
3
2
cot
1
3
2
3
2
>
−−
<
−−
x
se
xhgarc
x
se
xhtgarc
 
26. ∫ +162x
x
e
dxe
 
c
e
tgarc
x
+=
44
1
 
Considerando-se: 
dxedu
eu
eu
x
x
x
2
22
=
=
=
 
27. ∫
−
+ dx
x
x
1
3
 
.
32
32ln232
2
2ln22
2
1
2
1
ln
2
12.22
2
1
22
44
4
482
4
42
4
412
4
22.
13
22
222
2
2
c
x
x
xc
u
u
u
c
u
u
u
u
du
u
u
du
u
u
du
udu
u
du
u
uduu
u
u
+
+−
++
−+=+
−
+
−=
+
−
+
−=






−
−=
−
−
+=




−
+=





−
+=
−
=
−−
=
∫∫
∫∫∫∫
 
 
Considerando-se: 
duudx
ux
xu
2
3
3
2
2
=
−=
+=
 
 
28. ∫ xx
dx
3ln
3
2 
mathe
Realce
mathe
Realce
 458 
( ) ( ) .
3ln
3
1
3ln333ln 2 c
x
c
x
x
dx
x +
−
=+
−
== ∫
−
 
Considerando-se: 
dx
x
du
xu
3
3
3ln
=
=
 
29. ∫ + dxxsen )2cos4( pi 
( ) .2cos4cos
4
12cos4 cxxdxdxxsen ++−=+= ∫ ∫ pipi 
30. ∫
+ dxxx 1
2
2 
.
2ln
2
2ln
2
2
1
22 1
cc
xx
+=+=
+
 
Considerando-se: 
dxxdu
xu
2
12
=
+=
 
31. ∫ dxex
x23
 
ce x +=
23
6
1
 
Considerando-se: 
dxxdu
xu
6
3 2
=
=
 
32. ∫ + 2)2( t
dt
 
( )∫ −+= 22 t ctc
t
+
+
−
=+
−
+
=
−
2
1
1
)2( 1
. 
Considerando-se: 
dtdu
tu
=
+= 2
 
mathe
Realce
mathe
Realce
 459 
33. ∫ tt
dt
ln
 
.lnln ct += 
Considerando-se: 
t
dtdu
tu
=
= ln
 
34. ∫ − dxxx
2218 
( ) ( ) .21
3
4
2
3
21
4
18 2
32
3
2
2
cxc
x
+−
−
=+
−−
= 
Considerando-se: 
dxxdu
xu
4
21 2
−=
−=
 
35. ∫ + dxee
xx 252 )2( 
( ) ( ) .2
12
1
6
2
2
1 62
62
cec
e x
x
++=+
+
= 
Considerando-se: 
dxedu
eu
x
x
2
2
2
2
=
+=
 
36. ∫
+ 54
4
2t
dtt
 
 ( ) dttt 454 212 −∫ −= 
 
( )
.54
2
1
54
2
1 22 2
1
ctc
t
++=+
+
= 
Considerando-se: 
dttdu
tu
8
54 2
=
+=
 
mathe
Realce
mathe
Realce
 460 
37. ∫
−
dx
xsen
x
3
cos
 
cxsen +−−= |3|ln 
Considerando-se: 
dxxdu
xsenu
cos
3
−=
−=
 
38. ∫ + 5)1( vv
dv
 
( )
( ) cv
c
v
+
+
−=
+
−
+=
−
4
4
12
1
4
12
 
Considerando-se: 
dv
v
du
vu
2
1
1
=
+=
 
39. ∫ + dxxx 1
2
 
Considerando-se: 
duudxux
ux
21
1
2
2
=⇒−=
=+
 
( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( ) .11
3
211
5
411
7
2
1
3
21
5
41
7
2
3
2
5
4
7
2242
212211
23
357
357
246
224222
cxxxxxx
cxxx
c
uuuduuuu
duuuuduuuudxxx
++++++−++=
++++−+=
++−=+−=
+−=−=+
∫
∫∫∫
 
40. ∫
− dxex x
54
 
mathe
Realce
mathe
Realce
 461 
ce x +
−
=
−
5
5
1
 
Considerando-se: 
4
5
5xdu
xu
−=
−=
 
41. ∫ dttt
2cos 
ctsen += 2
2
1
, utilizando: 
tdtdu
tu
2
2
=
=
 
 
42. ∫ + dxxx 568
32
 
( ) ( ) ( ) .56
27
856
3
2
9
4
2
3
56
18
18 2
3
2
32
3
33
3
cxcxc
x
++=++=+
+
= 
Considerando-se: 
dxxdu
xu
2
3
18
56
=
+=
 
43. ∫ θθθ dsen 2cos2
2/1
 
( ) ( ) csencsen +=+= 2/32
3
1
2
3
2
2
1 2
3
θθ . 
Considerando-se: 
θθ
θ
ddu
senu
2cos2
2
=
=
 
44. ∫ + dxx )35(sec2 
cxtg ++= )35(
5
1
. 
Considerando-se: 
mathe
Realce
mathe
Realce
 462 
dxdu
xu
5
35
=
+=
 
45. ∫
−
3)cos5( θ
θθ dsen
 
( )
c+
−
−
=
−
2
cos5 2θ
. 
Considerando-se: 
θθ
θ
dsendu
u
=
−= cos5
 
46. ∫ duugcot 
cusendu
usen
u
+== ∫ ||ln
 
cos
 
Considerando-se: duudu
usenu
cos=
=
 
47. ∫ >+
−− 0,)1( 2/3 adtee atat 
( ) ( ) .1
5
2
2
5
11 252
5
ce
a
c
e
a
at
at
++−=+
+−
=
−
−
 
Considerando-se: 
( )dtaedu
eu
at
at
−=
+=
−
−1
 
48. ∫ dx
x
xcos
 
cxsen += 2 . 
Considerando-se: 
dx
x
du
xu
2
1
=
=
 
mathe
Realce
 463 
49. ∫ − dttt 4 
( ) ( )
( ) ( )
( ) ( ) ctttt
ctt
c
uuduuuduuuu
+−−+−−=
+−+−=
++=+=+= ∫∫
44
3
844
5
2
4
3
84
5
2
3
8
5
2822.4
2
35
35
242
 
Considerando-se: 
duudtut
ut
24
4
2
2
=⇒+=
=−
 
50. ∫ + dxxxsenx )42( 32 
cxxc
x
x
xdxxdxxsenx ++−=++−=+= ∫∫
43
4
3332 2cos
6
142cos
6
142 , 
sendo que na primeira integral usamos: 
dxxdu
xu
2
3
6
2
=
=
 
 
mathe
Realce