lista 02 calc 02A 2013
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lista 02 calc 02A 2013


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Lista 2 Ca´lculo II - A 2010-2 4
Universidade Federal Fluminense
EGM - Instituto de Matema´tica
GMA - Departamento de Matema´tica Aplicada
LISTA 2 - 2010-2
Integral indefinida
Integrac¸a\u2dco por substituic¸a\u2dco
Calcule as integrais dos exerc´\u131cios 1 a 24.
1.
\u222b
sen 3x cos 3x dx
2.
\u222b
sen \u3b8 cos3 \u3b8 d\u3b8
3.
\u222b
arctanx
1 + x2
dx
4.
\u222b
dx\u221a
x (1 +
\u221a
x)2
5.
\u222b
dx
4 + 3x2
6.
\u222b
x\u221a
1\u2212 x4 dx
7.
\u222b
y
(3y \u2212 4)3 dy
8.
\u222b
dt
t2 + 2t+ 2
9.
\u222b
x\u221a
x\u2212 1 dx
10.
\u222b
x(1 + x)
4
3 dx
11.
\u222b
cosx
4 + sen 2x
dx
12.
\u222b
tan2 x dx
13.
\u222b
sen 2x
3 + cos 2x
dx
14.
\u222b
dx
x ln
\u221a
x
15.
\u222b
3xex dx
16.
\u222b
ex\u221a
1\u2212 e2x dx
17.
\u222b
secx tanx dx
18.
\u222b
tanx dx
19.
\u222b
cotx dx
20.
\u222b
ex
cos2 (ex \u2212 2) dx
21.
\u222b
sen
\u221a
x\u221a
x
\u221a
cos3
\u221a
x
dx
22.
\u222b
18 tan2 x sec2 x
(2 + tan3 x)2
dx
23.
\u222b
cos(lnx)
x
dx
24.
\u222b
dx\u221a
1\u2212 4x2
25. Encontre a expressa\u2dco que define a func¸a\u2dco f , cujo gra´fico conte´m o ponto
(
0, 83
)
e cuja derivada
e´ f \u2032(x) = x
\u221a
1\u2212 x2.
Resolva os problemas de valor inicial dos exerc´\u131cios 26 a 29.
26.
\uf8f1\uf8f2\uf8f3
dy
dx
=
x\u221a
2x2 + 1
y(0) = 1
27.
{
y\u2032 =
x
2x2 + e2
y(0) = 1
28.
\uf8f1\uf8f2\uf8f3 dydx = e
1/x
x2
y(1) = 0
29.
{
f \u2032(x) =
(
1\u2212 sen 2x) sen 2x
f
(
pi
2
)
= 0
Resolva as integrais definidas dos exerc´\u131cios 30 a 37.
30.
\u222b 3
2
x\u221a
x\u2212 1 dx
31.
\u222b 2
1
ex
ex + e
dx
32.
\u222b \u221alnpi
0
2xex
2
cos
(
ex
2
)
dx
33.
\u222b e
1
dx
x
(
1 + ln2 x
) dx
34.
\u222b 1
2
0
x\u221a
1\u2212 x4 dx
35.
\u222b pi
2
0
e senx cosx dx
36.
\u222b pi
4
0
(
1 + etanx
)
sec2 x dx
37.
\u222b ln pi
2
ln pi
6
2ex cos (ex) dx
Lista 2 Ca´lculo II - A 2010-2 5
RESPOSTAS
1.
1
6
( sen 3x)2 + C
2. \u2212cos
4 \u3b8
4
+ C
3.
1
2
(arctanx)2 + C
4.
\u22122
1 +
\u221a
x
+ C
5.
\u221a
3
6
arctan
\u221a
3 x
2
+ C
6.
1
2
arcsenx2 + C
7.
2\u2212 3y
9(3y \u2212 4)2 + C
8. arctan (t+ 1) + C
9.
2
3
\u221a
(x\u2212 1)3 + 2\u221ax\u2212 1 + C
10.
3(1 + x)
10
3
10
\u2212 3(1 + x)
7
3
7
+ C
11.
1
2
arctan
(
1
2
senx
)
+ C
12. \u2212x+ tanx+ C
13. \u22121
2
ln |3 + cos 2x|+ C
14. 2 ln |ln\u221ax|+ C
15.
3xex
1 + ln 3
+ C
16. arcsen ex + C
17. secx+ C
18. ln | secx|+ C
19. \u2212 ln | cscx|+ C
20. tan (ex \u2212 2) + C
21. 4 (cos
\u221a
x)\u2212
1
2 + C
22. \u2212 6
2 + tan3 x
+ C
23. sen (lnx) + C
24.
1
2
arcsen (2x) + C
25. f(x) = \u22121
3
\u221a
(1\u2212 x2)3 + 3
26. y =
1
2
\u221a
2x2 + 1 +
1
2
27. y =
1
4
ln
(
2x2 + e2
)
+
1
2
28. y = \u2212e 1x + e
29. f(x) = sen 2x\u2212 1
2
sen 4x\u2212 1
2
30.
10
\u221a
2\u2212 8
3
31. ln
(
e+ 1
2
)
32. \u2212 sen (1)
33.
pi
4
34.
1
2
arcsen
(
1
4
)
35. e\u22121
36. e
37. 1