Engenharia

UNIVERSIDADE PAULISTA “UNIP” UNIVERSIDADE PAULISTA “UNIP”
Disciplina: Cálculo de Funções de Várias Variáveis Engenharia Básico Profª Juliana Brassolatti Gonçalves

Lista 5

1. Calcule as integrais indefinidas abaixo, usando as propriedades: a) d)

∫ 6 x dx
∫ − x 3 dx
2

b) e) h)

∫ (x

7

− 6 x + 8) dx

c) f)

∫ − 3x
⎜ ∫⎜ ⎝ ⎛

−4

dx
1 ⎞ ⎟ dx ⎟ x⎠

∫2
4

3

x dx

x+

g) 18 ⋅ e 6x dx

∫

∫ x dx

i) - 8 ⋅ cosec 2 x dx

∫

2. Calcule as integrais indefinidas abaixo, usando o método da Substituição: a)

∫

3 - 2x dx

b)

∫

1 5x + 4

dx

c)

∫x⋅

4

1 - x 2 dx

d) e) h)

2 ∫ 3x ⋅ 7 - 3x dx

e) f) i)

∫

2⋅e

x

2 ∫ sec ⋅ (3x + 2) dx

∫ (2 + senx )3 dx ∫ (2 x
2

dx x 6 ⋅ cos x

f) g)

∫ cos(3x + 4) dx
∫ x 2 + 4 dx x ∫ 2 - cosx dx ∫ x.sen5x dx ∫x
2

senx

+ 2x − 3

)

10

⋅ (2x + 1) dx

3. Calcule as integrais indefinidas abaixo, usando o método da Integração por Partes: a) b)

∫x⋅e

4x

dx
2

c)

∫ (x + 1) ⋅ cos2x dx

d)

⋅ lnx dx

e)

∫ ( x − 1) sec

x dx

4. Calcule as integrais definidas:

a)

−2

∫ (2 x + 5) dx
∫ (3 sec
0
2

0

⎛ x3 ⎞ ⎜ 3 x − ⎟ dx b) ∫ ⎜ 4 ⎟ ⎠ 0⎝
4

π

c)

∫ (1 + cos x) dx
0

π 3

π 2
2

d)

x) dx

e)

∫
0

1 + cos 2 x dx 2

π 2

f)

−π 2

∫ (8 x

2

+ senx) dx

g)

x ∫ xe dx
0

1

ln 2

h)

∫e
0

3x

dx

GABARITO 1. a) R: 3x + C
2

x8 b) R: − 3x 2 + 8 x + C 8
e) R: x3 + C

c) R:

1 +C x3

d) R:

1 +C x2

f) R:

2 3 x +2 x +C 3

g) R: 3 ⋅ e 6 x + C

h) R: 4 ⋅ ln x + C

i) R: 8 ⋅ cot gx + C

1 2. a) R: - ⋅ (3 − 2 x) 3 + C 3 1 d) R: - ⋅ (7 − 3x 2 ) 3 + C 3
e) R:

b) R:

2 ⋅ 5x + 4 + C 5 x 2 c) R: - ⋅ 4 (1 − x 2 ) 5 + C 5
f) R:

e) R: 4 ⋅ e

+C

1 ⋅ sen(3 x + 4) + C 3

1 ⋅ tg (3x + 2) + C 3

f) R:

−3 +C (2 + senx) 2

g) R:

1 ⋅ ln( x 2 + 4) + C 2

h) R: ln(2 − cos x) + C

i) R:

1 (2 x 2 + 2 x −