Gabarito parcial Lista de exercícios 11

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Universidade Federal de Santa Catarina
Centro de Cieˆncias F´ısicas e Matema´ticas
Departamento de Matema´tica
MTM3101 - Ca´lculo 1
Gabarito parcial da 11a lista de exerc´ıcios (06/11/2017 a 10/11/2017)
1. (a)
1
2
(
x
√
1 + x2 + ln
(
x+
√
1 + x2
))
+ C(b)
1
2
(
x
√
x2 − 1− ln
(
x+
√
x2 − 1
))
+ C(c)
arcsen (2x)
4
+
x
√
1− 4x2
2
+ C(d)
ln
(
x
2
+
√
1 +
x2
4
)
+ C(e)
−
√
1− y2 + C(f)
(g)
9
2
arcsen(t− 1
3
)
+
(
t− 1
3
)√
1−
(
t− 1
3
)2 + C(h)
1
2
arctg
x
2
+ C(i)
2
arcsen(x− 1
2
)
+
(
x− 1
2
)√
1−
(
x− 1
2
)2 + C(j)
2. 3 ln(x− 3)− 2 ln(x− 2) + C(a)
3
10
ln(x− 2) + 2
15
ln(x+ 3)− 1
6
lnx+ C(b)
1
2
ln(x+ 1)− 1
2
ln(x− 1)− 1
x− 1 + C(c)
5
4
lnx− 5
4
ln(x− 2)− 7
2(x− 2) + C(d)
2
5
lnx+
3
10
arctan
(
x+ 1
2
)
+
2
5
ln
(
2√
x2 + 2x+ 5
)
+ C(e)
(f)
lnx+ ln(x− 2)− 1
2
ln
(
2
x2 + 4x+ 4
)
−
√
2 arctan
(
x+ 2√
2
)
+ C(g)
x− ln(x2 + 4x+ 13)− 2 arctg
(
x+ 2
3
)
+C(h)
1
3. 1
4
(ln(2 + sen θ)− ln(2− sen θ)) + C(a) ln
∣∣∣tg x
2
∣∣∣− ln ∣∣∣1 + tg x
2
∣∣∣+ C(b)
(c) ln(3 cosx+ 2)− ln(cosx) + C(d)
2√
3
arctg
(
2 tg(x/2) + 1√
3
)
+ C(e) (f)
4.
− ln
∣∣∣∣ 1x +
√
1 +
1
x2
∣∣∣∣+C(a)
(b)
2 ln |x− 1|+ 1
2
ln(x2 + 2x+ 3) +
1√
2
arctg
x+ 1√
2
+ C(c)
−e
−x
10
sen(3x)− 3
10
e−x cos(3x) + C(d)
5
2
x2 + 5x+ 6 ln(x− 1) + C(e)
(f)
2
3
(g)
− ln | cos θ|+ cos2 θ − cos
4 θ
4
+ C(h)
√
x2 + 2x− 3− arccos
(
2
x+ 1
)
+C(i)
(j)
ln |2 secx+ 3|+ C(k)
e−pi/2 − 9
5
+ pi(l)
ln(x− 1)− 4
x− 1 + C(m)
(n)
x2
2
arctanx− x
2
+
arctanx
2
+ C(o)
1 + x2
4
(ln(1 + x2)− 1) + C(p)
(q)
x tanx+ ln(cosx)− x
2
2
+ C(r)
x+
19
4
ln(x− 3) + 1
4
ln(x+ 1) + C(s)
1
2(x− 1)2 −
2
x− 1 + C(t)
2