calc1lista11 - grazielle ufscar

Prévia do material em texto

089109 - Ca´lculo 1 - Turma C
De´cima Primeira lista de exerc´ıcios
Profa. Grazielle Feliciani Barbosa 30 de janeiro de 2017
Exerc´ıcio 1. Calcule
(a)
∫ 1
0
xex
2
dx (b)
∫ 0
−1
x(2x+ 1)50dx (c)
∫ 1
0
x
(x2 + 1)5
dx
(d)
∫ 1
−1
x4(x5 + 3)3dx (e)
∫ pi
2
pi
6
senx(1− cos2 x)dx (f)
∫ pi
3
0
sen 3xdx
(Respostas: (a)
1
2
e− 1
2
(b) − 1
102
(c)
15
128
(d) 12 (e)
3
8
√
3 (f)
5
24
)
Exerc´ıcio 2. Calcule
(a)
∫
x3 cosx4dx (b)
∫
sen 5x cosxdx (c)
∫
tgx sec2 xdx
(d)
∫
sec2 x
3 + 2 tgx
dx (e)
∫ (
5
x− 1 +
2
x
)
dx; (f)
∫
1
a2 + x2
dx
(g)
∫
1
x lnx
dx (h)
∫
1
x
cos(lnx)dx
(Respostas: (a)
1
4
senx4 + k (b)
1
6
sen 6x + k (c)
1
2
tg 2x + k (d)
1
2
ln |3 + 2 tgx| + k (e)
5 ln |x− 1|+ 2 ln |x|+ k (f) 1
a
arctg
x
a
+ k (g) ln | lnx|+ k (h) sen (lnx) + k).
Exerc´ıcio 3. Seja f uma func¸a˜o par e cont´ınua em [−r, r], com r > 0.
(a) Mostre que
∫ 0
−r
f(x)dx =
∫ r
0
f(x)dx.
(b) Conclua de (a) que
∫ r
−r
f(x)dx = 2
∫ r
0
f(x)dx. Interprete graficamente.
Exerc´ıcio 4. Seja f uma func¸a˜o ı´mpar e cont´ınua em [−r, r], r > 0. Mostre que∫ r
−r
f(x)dx = 0.
Exerc´ıcio 5. Suponha f cont´ınua em [0, 4]. Calcule
∫ 2
−2
xf(x2)dx.
1
Exerc´ıcio 6. Calcule
(a)
∫ 2
0
x2
(x+ 1)2
dx (b)
∫ 2
−2
x2(x3 + 3)10dx (c)
∫ pi
6
0
senx cos2 xdx
(d)
∫ pi
4
0
cosx sen 5xdx.
(Respostas: (a)
8
3
− 2 ln 3 (b) 1168 (c) −1
8
√
3 +
1
3
(d)
1
48
).
Exerc´ıcio 7. Calcule:
(a)
∫
xe−x dx (b)
∫
x cos x dx (c)
∫
[x2ex
3 − x3 lnx] dx
(d)
∫
arcsen 2x dx (e)
∫
x2exdx (f)
∫
x2 lnxdx
(g)
∫
(lnx)2dx (h)
∫
ex cosxdx (i)
∫
x2 senxdx
(Respostas: (a) −e−x(x+ 1) + k (b) x senx+ cosx+ k (c) e
x3
3
− x
4 lnx
2
+ k (d) x arcsen 2x+
1
2
√
1− 4x2 + k (e) ex(x2− 2x+ 2) + k (f) 13x3(lnx− 13) + k (g) x(lnx)2− 2x(lnx− 1) + k (h)
1
2e
x( senx+ cosx) + k (i) −x2 cosx+ 2x senx+ 2 cosx+ k)
Exerc´ıcio 8. Calcule
∫
e−st sen tdt, onde s > 0 e´ constante.
(Resposta: − e
−st
1 + s2
(cos t+ s sen t) + k
Exerc´ıcio 9. Para todo n ≥ 1 e todo s > 0, verifique que∫
tne−stdt = −1
s
tne−st +
n
s
∫
tn−1e−stdt.
Exerc´ıcio 10. Calcule
(a)
∫ 1
0
xexdx (b)
∫ 2
1
lnxdx
(c)
∫ pi
2
0
ex cosxdx (d)
∫ x
0
t2e−stdt (s 6= 0)
(Respostas: (a) 1 (b) 2 ln 2− 1 (c) 1
2
(e
pi
2 − 1) (d) −1
s
x2e−sx − 2
s2
xe−sx − 2
s3
e−sx +
2
s3
)
Exerc´ıcio 11. Sejam m e n naturais na˜o nulos. Verifique que∫ 1
0
xn(1− x)mdx = m
n+ 1
∫ 1
0
xn+1(1− x)m−1dx.
2
Exerc´ıcio 12. Calcule
∫ √
a2 + x2 dx, a > 0.
(Resposta:
1
2
[
x
a2
√
a2 + x2 + ln
∣∣∣∣∣x+
√
a2 + x2
a
∣∣∣∣∣
]
+ k)
Exerc´ıcio 13. Deduza a a´rea do c´ırculo de raio r, r > 0.
(Resposta: pir2)
Exerc´ıcio 14. Calcule
(a)
∫ √
−x2 + 2x+ 3 dx (b)
∫ √
6− 3x2 dx (c)
∫
1
x
√
1 + x2
dx
(Respostas: (a) 2 arcsen
x− 1
2
+
x− 1
2
√
4− (x− 1)2 + k (b) 1
2
x
√
(6− 3x2) +√3 arcsen 1
2
√
2x+ k
(c) ln
∣∣∣∣ x1 +√1 + x2
∣∣∣∣+ k).
Exerc´ıcio 15. Calcule a a´rea da elipse descrita pela equac¸a˜o 9x2 + y2 ≤ 3.
(Resposta: pi).
Exerc´ıcio 16. Sejam m e n constantes na˜o nulas Mostre que:∫
mx+ n
1 + x2
dx =
m
2
ln(1 + x2) + n arctgx+ k
Exerc´ıcio 17. Calcule
(a)
∫
x3 + x+ 1
x2 − 4x+ 3 dx (b)
∫
3
x2 + 3
dx
(c)
∫
x2 + 3
x2 − 9 dx (d)
∫
x2 + 1
(x− 3)2 dx
(Respostas: (a)
x2
2
+ 4x− 3
2
ln |x− 1|+ 31
2
ln |x− 3|+ k (c) x+ 2 ln |x− 3| − 2 ln |x+ 3|+ k)
Exerc´ıcio 18. Calcule e verifique o resultado por derivac¸a˜o.
(a)
∫
x+ 3
x(x− 3)(x− 4) dx (b)
∫
3
x3 − 16x dx
(c)
∫
x3 + 1
x3 − x2 − 2x dx (d)
∫
5
(x2 − 1)(x2 − 9) dx
Exerc´ıcio 19. Siga as instruc¸o˜es:
(a) Determine A, B, C e D tais que
x− 3
(x− 1)2(x+ 2)2 =
A
x− 1 +
B
(x− 1)2 +
C
x+ 2
+
D
(x+ 2)2
.
3
(b) Calcule
∫
x− 3
(x− 1)2(x+ 2)2 dx.
(Respostas:
7
27
ln |x− 1|+ 6
27(x− 1) −
7
27
ln |x+ 2|+ 15
27(x+ 2)
+ k.)
Exerc´ıcio 20. Calcule as integrais:
(a)
∫
2x− 1
(x− 1)(x− 2) dx (b)
∫
x
(x+ 1)(x+ 3)(x+ 5)
dx
(c)
∫
x4
(x2 − 1)(x+ 2) dx (d)
∫
dx
(x− 1)2(x− 2)
(e)
∫
x− 8
x3 − 4x2 + 4x dx (f)
∫
dx
x(x2 + 1)
dx
(g)
∫
dx
x3 + 1
(h)
∫
4x2 − 8x
(x− 1)2(x2 + 1)2 dx.
(Respostas: (a) ln
∣∣∣∣(x− 2)3x− 1
∣∣∣∣+ k (b) 18 ln
∣∣∣∣ (x+ 3)6(x+ 5)5(x+ 1)
∣∣∣∣+ k
(c)
x2
2
− 2x+ 1
6
∣∣∣∣ x− 1(x+ 1)3
∣∣∣∣+ 163 ln |x+ 2|+ k (d) 1x− 1 + ln
∣∣∣∣x− 2x− 1
∣∣∣∣+ k
(e)
3
x− 8 + ln
(x− 2)2
x2
+ k (f) ln
|x|√
x2 + 1
+ k (g)
1
6
ln
(x+ 1)2
x2 − x+ 1 +
1√
3
arctg
2x− 1√
3
+ k
(h)
3x2 − 1
(x− 1)(x2 + 1) + ln
(x− 1)2
x2 + 1
+ arctgx+ k )
Exerc´ıcio 21. Calcule as integrais:
(a)
∫ √
a2 − x2
x2
dx (b)
∫
dx
x2
√
1 + x2
dx
(c)
∫ √
x2 − a2
x
dx (d)
∫
1√
(a2 + x2)3
dx.
(Respostas: (a) −
√
a2 − x2
x
− arcsen x
a
+ k (b) −
√
1 + x2
x
+ k (c)
√
x2 − a2 − a arccos a
x
+ k
(d)
x
a2
√
a2 + x2
+ k)
4