Calculo I - Lista de Exercícios Nº 10 - COM GABARITO

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Cálculo I - Lista de Exerćıcios no¯ 10 - 1
o
¯ semestre/2016
1. Calcule as integrais:
(a)
∫ 5√
x2 dx (b)
∫
(3x4 + 2x− 7)dx (c)
∫
(3 cos x− 7sen x)dx
(d)
∫√
x(x+ 1)dx (e)
∫√
x
(
x+ 1x
)
dx (f)
∫
5t2 + 6
t4/3
dt
(g)
∫
2
3
√
y
dy (h)
∫ (√
x+
1√
x
)
dx (i)
∫ (
2
u4
−
3
u3
+ 8
)
du
2. Calcule as integrais, usando substituição:
(a)
∫
xex
2
dx (b)
∫
(x− 2)5 dx (c)
∫√
2x+ 1 dx
(d)
∫
x2
(1+ x3)2
dx (e)
∫
x
(x+ 1)5
dx (f)
∫
x(x+ 1)100 dx
(g)
∫
x2√
x+ 1
dx (h)
∫
cos xsen 5xdx (i)
∫
sen xsen 2xdx
(j)
∫
e7x dx (k)
∫
cos2 xdx (l)
∫
tg xdx
(m)
∫
x+ sen 7xdx (n)
∫
x4 cos x5 dx (o)
∫
1
x2 + 4
dx
(p)
∫
ex√
1− ex
dx
3. Calcule as integrais, usando integração por partes:
(a)
∫
xex dx (b)
∫
x2ex dx (c)
∫
x3ex dx
(d)
∫
x ln xdx (e)
∫
ln xdx (f)
∫
4xe5x dx
(g)
∫
xsen xdx (h)
∫
x
√
x+ 2 dx (i)
∫
x2 ln xdx
(j)
∫
arcsen xdx (k)
∫
e2xsen xdx
4. Calcule as integrais, usando frações parciais:
(a)
∫
x+ 1
x2 − x− 2
dx (b)
∫
x
x2 + 4
dx (c)
∫
x
(x+ 3)2
dx
(d)
∫
x− 1
x2 + 4
dx (e)
∫
2x+ 3
x(x− 2)
dx (f)
∫
x2 + 3
x2 − 9
dx
(g)
∫
x− 1
x2 + 2x+ 10
dx (h)
∫
x
x2 − 4
dx (i)
∫
x
x(x+ 1)
dx
(j)
∫
1
(x+ 1)(x+ 2)(x+ 3)
dx (k)
∫
2x+ 1
(x+ 1)3(x2 + 4)2
dx
5. Calcule as integrais:
(a)
∫
xe−x
2
dx (b)
∫
(cos x)7 dx (c)
∫
(sen x)3 dx
(d)
∫
sec2 xtg xdx (e)
∫
(x+ 7)9 dx (f)
∫
x2sen 3xdx
(g)
∫
1
x
cos(ln x) dx (h)
∫
sec2 x
3+ 2tg x
dx (i)
∫
2x√
1− 4x2
dx
(j)
∫
1
x ln x
dx (k)
∫
cos x
4− sen 2x
dx (l)
∫
1
x2 + 4x+ 3
dx
(m)
∫
1
x2 + 2x+ 2
dx (n)
∫
cos
√
x√
x
dx (o)
∫
(ln x)3
x
dx
(p)
∫
1
x2 + 4x+ 8
dx (q)
∫ √
x+ 4
x
dx (r)
∫
cos(ln(x)) dx
Instituto de Matemática Universidade Federal de Mato Grosso do Sul
6. Calcule as integrais trigonométricas:
(a)
∫
sen 3(x) cos2(x)dx (b)
∫
sen 5(x) cos3(x)dx (c)
∫
sen 4(x) cos2(x)dx
(d)
∫
sen (5x)sen (2x)dx (e)
∫
sen (3x) cos(x)dx (f)
∫
tg 3(x) sec2(x)dx
(g)
∫
tg 3(x) sec5(x)dx (h)
∫
tg 2(t) sec4(t)dt (i)
∫
tg 2(x) sec3(x)dx
(j)
∫
cos(θ) cos5(sen (θ))dθ (k)
∫
cotg 3(y)cosec 3(y)dy (l)
∫
1
cos(x) − 1
dx
7. Calcule as integrais usando substituição inversa:
(a)
∫
1
t2
√
25− t2
dt (b)
∫
1√
x2 + 16
dx (c)
∫
1
x2
√
x2 − 9
dx
(d)
∫
x
√
1− x4dx (e)
∫
x arcsen (x)dx (f)
∫
1√
9x2 + 6x− 8
dx
(g)
∫√
1− (x− 1)2dx (h)
∫√
x− x2dx (i)
∫
arctg (ex)
ex
dx
8. Calcule as integrais:
(a)
∫
ex
1+ e2x
dx (b)
∫ √
1+ lny
y
dy (c)
∫
1
1+
√
x
dx
(d)
∫
sen (x) + sec(x)
tg (x)
dx (e)
∫
earctg (z)
1+ z2
dz (f)
∫
t4 ln(t)dt
(g)
∫
x sen 2(x)dx (h)
∫
ex+e
x
dx (i)
∫√
1+ x
1− x
dx
(j)
∫
1√
x− 3
√
x
dx (k)
∫
1
3
√
x+ 4
√
x
dx (l)
∫
cos(x)
4− sen 2(x)
dx
(m)
∫
1
1− cos(x) + sen (x)
dx (n)
∫
1
sen (x) + cos(x)
dx (o)
∫
sen (2x)
1+ cos(x)
dx
9. Calcule as integrais definidas.
(a)
∫ 1
0
2xdx (b)
∫ e
1
dx
x
(c)
∫ 1
0
e2xdx (d)
∫ 5
0
dx
x+ 1
(e)
∫ 0
5
dx
x+ 1
(f)
∫ π
2
−π
2
sen x dx (g)
∫x
0
dt
1− t
(h)
∫ 2
−1
|x− x2|dx (i)
∫ 2
1
ln x dx (j)
∫1
0
√
1− x2dx.
10. Esboce e encontre a área da região limitada pelas curvas dadas.
(a) y = x+ 1, y = 9− x2, x = −1 e x = 2.
(b) y = sen x, y = cos x, x = 0 e x =
π
2
.
(c) y = |x| e y = x2 − 2.
11. Calcule a integral, interpretando-a como área de uma região.∫ 2
0
1−
√
1− (x− 1)2dx.
Instituto de Matemática Universidade Federal de Mato Grosso do Sul
Cálculo I - Lista de Exerćıcios no¯ 10 - Gabarito - 1
o
¯ semestre/2016
1. (a) 7
5
5
√
x7 + C (b) −7x+ x2 + 3x
5
5
+ C (c) 3sen x+ 7 cos x+ C
(d) 2
15
x3/2(5+ 3x) + C (e) 2
5
√
x(5+ x2) + C (f) 3(−6+t
2)
t1/3
+ C
(g) 4
3
√
y+ C (h) 2
3
√
x(3+ x) + C (i) −4+9u+48u
4
6u3
+ C
2. (a) e
x2
2
+ C (b) 1
6
(x− 2)6 (c) 1
3
(1+ 2x)3/2 + C
(d) −1
3(1+x3)
+ C (e) − 4x+1
12(x+1)4
+ C (f) (x+1)
102
102
− (x+1)
101
101
+ C
(g)
2
15
√
x+ 1(8− 4x+ 3x2) + C (h) 1
6
sen 6x+ C (i) −cosx+ cos
3x
3
+ C
(j) e
7x
7
+ C (k) 1
2
(x− sen (2x)
2
) + C (l) − ln(cos x) + C
(m) x
2
2
+ 1−cosx
7
+ C (n) senx
5
5
+ C (o) arctg (x
2
) + C
(p) −2
√
1− ex + C
3. (a) ex(−1+ x) + C (b) ex(2− 2x+ x2) + C (c) ex(−6+ 6x− 3x2 + x3) + C
(d) 1
4
x2(−1+ 2 ln(x)) + C (e) x(−1+ ln(x)) + C (f) 4
25
e5x(−1+ 5x) + C
(g) −x cos(x) + sen (x) + C (h) 2
15
(2+ x)3/2(−4+ 3x) + C (i) 1
9
x3(−1+ 3 ln(x)) + C
(j)
√
(1− x2) + xarcsen (x) + C (k) −1
5
e2x(cos(x) − 2sen (x)) + C
4. (a) ln(−2+ x) + C (b) 1
2
ln(4+ x2) + C (c) 3
x+3 + ln(x+ 3) + C
(d) 1
2
arctg (x
2
) + ln(4+ x2)) + C (e) 7
2
ln(2− x) − 1
2
(3 ln(x)) + C (f) x+ 2 ln(3− x) − 2 ln(3+ x) + C
(g) 1
6
(−4arctg 1+x
3
+ 3 ln(10+ 2x+ x2)) + C (h) 1
2
ln(−4+ x2) + C (i) ln(1+ x) + C
(j) − ln(2+ x) + 1
2
ln(3+ 4x+ x2) + C
(k)
200
(1+x)2
− 480
(1+x
+
(840−190x)
(4+x2)
−31arctg (x
2
)+608 ln(1+x)−304 ln(4+x2))
10000
+ C
5. (a) −e
−x2
2
+ C (b) sen x− sen 3x+ 3
5
sen 5x− 1
7
sen 7x+ C (c) − cos x+ cos
3x
3
+ C
(d) sec
2 x
2
+ C (e) (x+7)
10
10
+ C (f) − 1
27
(−2+ 9x2) cos(3x) + 2
9
xsen (3x) + C
(g) sen (ln x)C (h) ln(3+ xtg x)2 + C (i) −1
2
√
1− 4x2 + C
(j) x ln(ln x) + C (k) 1
4
(− ln(2− sen (x)) + ln(2+ sen (x))) + C (l) 1
2
(ln(1+ x) − ln(3+ x)) + C
(m) arctg (1+ x) + C (n) 2sen (
√
x) + C (o) (lnx)
4
4
+ C
(p) 1
2
arctg ((2+ x)/2) + C (q) 2(
√
(4+ x) + ln(2−
√
(4+ x)) − ln(2+
√
(4+ x))) + C
(r) 1
2
x(cos(ln(x)) + sen (ln(x))) + C
6. (a) − cos
3 x
3
+ cos
5x
5
+ C (b) sen
6x
6
− sen
8x
8
+ C (c) x
16
− sen (4x)
64
− sen
3(2x)
48
+ C
(d) sen (3x)
6
− sen (7x)
14
+ C (e) − cos(2x)
4
− cos(4x)
8
+ C (f) tg
4x
4
+ C
(g) sec
7 x
7
− sec
5 x
5
+ C (h) tg
5t
5
+ tg
3x
3
+ C
(i) 1
16
(
2(sen 3x+senx)
(1−sen 2x)2
+ ln |1− sen x|+ ln |1+ sen x|
)
+ C
(j) sen (sen θ) − 2sen
3(senθ)
3
+ sen
5(senθ)
5
+ C (k) 1
3
y3cotg 3(x)cosec 3(x) + C (l) cot(x/2) + C
7. (a) −
√
25−t2
25t
+ C (b) arcsenh (x
4
) + C
(c)
√
−9+x2
9x
+ C (d) 1
4
(x2
√
1− x4 + arcsen (x2)) + C
(e) 1
4
(x
√
1− x2 + (−1+ 2x2)arcsen (x)) + C (f) ln(1+ 3x+
√
−8+ 6x+ 9x2)1/3 + C
(g)
√
−(−2+x)x(
√
−2+x(−1+x)
√
x−2 ln(
√
−2+x+
√
x))
2
√
−2+x
√
x
+C
(h)
√
−(−1+x)x(
√
−1+x
√
x(−1+2x)−ln(
√
−1+x+
√
x))
4
√
−1+x
√
x
+ C
(i) −e−xarctg (ex) − 1
2
ln(1+ e−2x) + C
Instituto de Matemática Universidade Federal de Mato Grosso do Sul
8. (a) arctg (ex) + C
(b) 2
3
(1+ ln(y))(3/2) + C
(c) 2
√
x− 2 ln(1+
√
x) + C
(d) − ln(cos(x/2)) + ln(sen (x/2)) + sen (x) + C
(e) earctg (z)
(f) 1
25
t5(−1+ 5 ln(t)) + C
(g) 1
8
(− cos(2x) + 2x(x− sen (2x))) + C
(h) ee
x
+ C
(i)
√
(1+x)/(1−x)((−1+x)
√
1+x+2
√
1−xarcsen (
√
1+x/
√
2))√
1+x
+ C
(j) 6x1/6 + 3x1/3 + 2
√
x+ 6 ln(1− x1/6) + C
(k) 3x
2/3
2
− 12x
7/12
7
− 12x
5/12
5
+
√
x+ 3x1/3 − 4x1/4 + 6x1/6 − 12x1/12 + 12 ln(x1/12 + 1) + C
(l) 1
4
(− ln(2− sen (x)) + ln(2+ sen (x))) + C
(m) ln(sen (x/2)) − ln(cos(x/2) + sen (x/2)) + C
(n) arcsen x−
√
1− x2 + C
(o) −2 cos x+ 4 ln(cos(x/2)) + C
9. Calcule as integrais definidas.
(a) 1 (b) 1 (c) 1
2
(e2 − 1) (d) ln(6) (e) − ln(6)
(f) 0 (g) − ln(1− x) (h) 11
6
(i) ln(4) − 1 (j) π
4
10. Esboce e encontre a área da região limitada pelas curvas dadas.
(a)
−4. −2. 2.
2.
4.
6.
8.
0
a
c
b d
Região de integração: Limitada acima pela
parábola, abaixo pela reta e lateralmente por
x = −1 e x = 2. R = 39
2
.
(b)
−1−0.5 0.5 1 1.5 2. 2.5
−1
0.5
1
1.5
2
2.5
0
ba
Região de integração: No intervalo de 0 à π
4
, a
função cosseno é a limitante superior e a seno
a inferior. De π
4
à π
2
, a função seno é a limi-
tante superior e a cosseno a inferior. Dos lados
é limitada por x=0 e x=π
2
. R =2
√
2− 2.
(c)
−3. −2. −1. 1. 2.
−3.
−2.
−1.
1.
2.
0
f
c
A B
Região de integração: Limitada superiormente
pela função módulo de x e inferiormente pela
parábola, no intervalo de −2 à 2. R = 20
3
.
11. 2−
π
2
Instituto de Matemática Universidade Federal de Mato Grosso do Sul