Lista 4 de Cálculo I

Prévia do material em texto

4a Lista de Ca´lculo I
Ca´lculo de a´reas
1. Calcule a a´rea da regia˜o do plano limitada pelas curvas:
(a) y = x3 e y = x2. (Obs.: Lembre-se que se 0 ≤ x ≤ 1, enta˜o x3 ≤ x2)
(b) y = 1
x
, x = 1, x = 2 e o eixo x.
(c) y = 3− x2 e y = x2 − 1
(d) y = 2senx e y = tgx, sendo x ∈ [0, pi
2
]
(e) y = 3− x2, y = 2x e y = −2x.
(f) 2y2 = x + 4 e y2 = x.
(g) x = 2y − y2 e y − x− 2 = 0.
(h) y = 1− x2, a reta tangente a essa para´bola no ponto de abscissa 1
2
e o eixo dos x.
Me´todos de Integrac¸a˜o
2. Calcule as integrais atrave´s do me´todo da substituic¸a˜o:
(a)
∫
xsen(2x2)dx (b)
∫ √
3− 2xdx (c)
∫
9x2√
1− x3dx
(d)
∫
1√
x(1 +
√
x)2
dx (e)
∫
x
√
4− xdx (f)
∫
x3
√
x2 + 1dx
(g)
∫ √
x− 1
x5
dx, x > 0 (h)
∫
1
[xlnx]2
dx (i)
∫ √3
0
4x√
x2 + 1
dx
3. Calcule as integrais usando o me´todo de integrac¸a˜o por partes.
(a)
∫
xlnxdx (b)
∫
ln(x2 + 1)dx (c)
∫
xarctg(x)dx
(d)
∫
x2e−xdx (e)
∫
e2xcos(3x)dx (f)
∫
sen(lnx)dx
4. Calcule as integrais usando o me´todo de frac¸o˜es parciais.
(a)
∫
2x− 1
(x− 1)(x− 2)dx (b)
∫
x
(x + 1)(x + 3)(x + 5)
dx (c)
∫
x− 8
x3 − 4x2 + 4xdx
(d)
∫
1
x3 + 1
dx (e)
∫
x2 + 2x + 1
(x2 + 1)2
dx (f)
∫
2x + 2
(x2 + 1)(x− 1)3dx
5. Calcule as integrais usando o me´todo de substituic¸a˜o trigonome´trica.
(a)
∫
1
x2
√
1 + x2
dx (b)
∫
x2
√
4− x2dx (c)
∫
1
x3
√
x2 − 9dx
GABARITO
1. (a) 1
12
u.a. (b)ln2 u.a. (c)
16
√
(2)
3
u.a. (d)(1− ln2)u.a. (e) 10
3
u.a. (f) 32
3
u.a. (g) 9
2
u.a.
(h) 7
96
u.a.
2.
(a) I = −cos(2x
2)
4
+ C (b) I =
−
√
(3−2x)3
3
+ C (c) I = −6√1− x3 + C
(d) I = − 2
1+
√
x
+ C (e) I = 2
5
√
(4− x)5 − 8
3
√
(4− x)3 + C
(f)
√
(x2+1)5
5
−
√
(x2+1)3
3
+ C (g) 2
3
√
(1− 1
x
)3 + C (h) I = − 1
lnx
+ C (i)4.
3.
(a) I = 1
2
x2
(
lnx− 1
2
)
+ C (b) I = xln(x2 + 1)− 2x + 2arctg(x) + C
(c)I = 1
2
[(x2+1)arctg(x)−x]+C (d)−(x2)2x+2)e−x+C (e)I = e2x
13
(3sen3x+2cos3x)+C
(f) I = 1
2
xsen(lnx)− cos(lnx) + C
To be continued...
4.
5.