Integrais - Parte 2

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Ca´lculo I - Lista de Exerc´ıcios no¯ 13 - 1
o
¯ semestre/2015
1. Calcule as integrais trigonome´tricas:
(a)
∫
sen 3(x) cos2(x)dx (b)
∫
sen 5(x) cos3(x)dx (c)
∫
sen 4(x) cos2 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)cossec 3(y)dy (l)
∫
1
cos(x) − 1
dx
2. Calcule as integrais usando substituic¸a˜o 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
3. 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)
∫
t4ln(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
4. Determine a soluc¸a˜o geral das seguintes equac¸o˜es diferenciais ordina´rias.
(a) y ′ = y (b) y ′ = y/x (c) y ′ = x2/y
(d) y ′ = xy3 (e) 2x+ 3yy ′ = 0 (f) x2y ′ = 1− y
5. Resolva os seguintes problemas de valor inicial.
(a) y ′ + y = 0, y(0) = 2 (b) xy ′ = 3y, y(2) = −8 (c) (x+ 1)y ′ = y, y(−2) = −3
(d) xy ′ + y = 0, y(1) = 1 (e) y ′ + 3xy = 0, y(0) = pi (f) (y+ 2)y ′ = sen x, y(0) = 0
UFMS / INMA Disciplina: Ca´lculo I Turmas: 1, 2, 3 e 7