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Disciplina: Cálculo II - MAT
Curso:
Prof.(a): Reinaldo Lima Data:
Aluno(a):
Lista de exercícios I - Técnicas de Integração
Questão 01. Usando substituição de variável, resolva as integrais abaixo :
(a)
Z
5 x dx (i)
Z
xdx √ 2 x^2 + 3
(r)
Z
(x + 1)dx
x^2 + 2x + 3
(b)
Z
sen (ax)dx, com a ̸= 0. (j)
Z
cotg x
sen 2 x
dx (s)
Z
dx
x ln x
(c)
Z
dx
sen 2 (3x − 1)
(l)
Z
dx
cos^2 x
tg x − 1
(t)
Z
x^2 +4x+ (x + 2)dx
(d)
Z
cos(5x)dx (m)
Z
ln(x + 1)
x + 1
dx (u)
Z
dx
1 + 2x^2
(e)
Z
dx
3 x − 7
(n)
Z
cos xdx √ 2sen x + 1
(v)
Z
dx √ 16 − 9 x^2
(f)
Z
tg (2x)dx (o)
Z
sen (2x)dx √ 1 + sen 2 x
(x)
Z
dx
4 − 9 x^2
(g)
Z
e x cotg (e x )dx (p)
Z
arcsen xdx √ 1 − x^2
(y)
Z
dx √ x^2 + 9
(h)
Z
x
x^2 + 1dx (q)
Z
arctg 2 x
1 + x^2
dx (z)
Z
arccos x − x √ 1 − x^2
dx
Questão 02. Use integração por partes para resolver as integrais abaixo:
(a)
Z
x 2
e x dx (f)
Z
x
sen 2 x
dx
(b)
Z
16 x 3
ln xdx (g)
Z
3 x 8 cos(x 3 )dx
(c)
Z
x 2
sen xdx (h)
Z
x 5
1 + 4e x^3
dx
(d)
Z
arctg (3x)dx (i)
Z
e
√ 2 x+ dx
(e)
Z
arcsen (x − 2)dx (j)
Z
xarctg x √ 1 + x^2
dx
Questão 03. Resolva as integrais contendo trinômio ax 2
(a)
Z
dx
x^2 + 2x + 5
(d)
Z
x + 3 √ 3 + 4x − 4 x^2
dx
(b)
Z
dx
x^2 − 6 x + 5
(e)
Z
(x + 5) √ 2 x^2 + 4x + 3
(c)
Z
(x + 5)dx
2 x^2 + 4x + 3
(f)
3 x + 3 p x(2x − 1)
dx
Questão 04. Resolva as integrais de funções racionais:
(a)
Z
(x + 1)
2 x + 1
dx (h)
Z
3 x − 7
x^3 + x^2 + 4x + 4
dx
(b)
Z
xdx
(x + 1)(x + 3)(x + 5)
dx (i)
Z
8 x − 16
16 − x^4
dx
(c)
Z
dx
(x − 1)^2 (x − 2)
(j)
Z
(x^2 − 2 x + 3)dx
(x^2 + 1)(x − 1)^2
(f)
Z
sen 2 (3x)dx (o)
Z
sen (5x)sen (3x)dx
(g)
Z
sen 2 (x) cos 2 (x)dx (p)
Z
sen (x) cos(5x)dx
(h)
Z
tg 3 xdx
Questão 07. Resolva as integrais usando substituição trigonométrica:
(a)
Z √
a^2 − x^2
x^2
dx (e)
Z
dx p (4 + x^2 )^5
(b)
Z
x 2
4 − x^2 dx (f)
Z
dx
(x + 1)^4
x^2 + 2x + 10
(c)
Z
dx
x^2
1 + x^2
(g)
Z
4 + x^2 dx
(d)
Z √
x^2 − a^2
x
dx (h)
Z
dx
(x + 1)^2
x^2 + 2x + 2
Respostas
Questão 01.
(a)
25 x
5 ln(2)
2 x^2 + 3 + C (r)
ln x 2
(b) −
cos(ax)
a
cotg 2 x
2
(c) −
cotg (3x − 1)
3
p tg x − 1 + C (t)
3 x
(^2) +4x+
2 ln 3
+ C
(d)
sen (5x)
5
ln 2 (x + 1)
2
arctg
2 x
+ C
(e)
ln | 3 x − 7 | + C (n)
2sen x + 1 + C (v)
arcsen
3 x
4
+ C
(f) −
ln | cos 2x| + C (o) 2
1 + sen 2 x + C (x)
ln
2 + 3x
2 − 3 x
+ C
(g) ln |sen (e x )| + C (p)
arcsen 2 x
2
x^2 + 9 + C
(h)
q (x^2 + 1)
3
arctg 3 x
3
arccos 2 (x) +
1 − x^2 + C
Questão 02.
(a) x 2 e x
- C (f) −xcotg x + ln |sen x| + C
(b) ln x
4 x 4
x 4
x 3
x 3
− 2sen
x 3
+ C
(c) −
x 2 − 1
cos x + 2xsen x + C h) e x^3
4 x^3 − 4
3
x^6
6
+ C
(d) xarctg (3x) −
ln
9 x 2
2 x + 1 − 1
e
√ 2 x+
(e) (x − 2)arcsen (x − 2) +
−x^2 + 4x − 3 + C (j)
1 + x^2 arctg x − ln x +
1 + x^2 + C
Questão 03.
(a)
arctg
x + 1
2
2 x^2 + 4x + 3 + 2
2 ln
2 x^2 + 4x + 3 +
2(x + 1) + C
(b)
ln
x − 5
x − 1
2 x^2 − x +
ln 4 x − 1 +
p 8 (2x^2 − x) + C
(c)
ln 2 x 2
2 arctg
h√ 2(x + 1)
i
(d) −
3 + 4x − 4 x^2 +
arcsen
2 x − 1
2
+ C
Questão 04.
(a)
x +
ln | 2 x + 1| + C (h) ln
x^2 + 4
(x + 1)^2
arctg
x
2
+ C
(b)
ln
(x + 3) 6
(x + 5)^5 (x + 1)
4 + x^2 − ln |2 + x| − arctg
x
2
+ C
(c)
x − 1
x − 2
x − 1
x^2 + 1 − ln |x − 1 | +
1 − x
+ C
(e)
cos^5 x +
√ (^3) cos x + C (n) − ln 1 − tg
x
2
+ C
(f)
x
2
sen (6x)
12
sen (2x) −
sen (8x)
4
+ C
(g)
x
8
sen (4x)
32
cos(6x)
12
cos(4x)
8
+ C
(h)
tg 2 x
2
Questão 07.
(a) −
a^2 − x^2
x
− arcsen
x
a
+ C
(b) 2arcsen
x
2
x
2
4 − x^2 +
x 3
4 − x^2 + C
(c) −
1 + x^2
x
+ C
(d)
x^2 − a^2 − (a) arccos
a
x
+ C
(e)
x √ 4 + x^2
x^3
3 (4 + x^2 )
4 + x^2
+ C
(f)
p 9 + (x + 1)^2
34 (x + 1)
q [9 + (x + 1)^2 ]
3
35 (x + 1)^3
+ C
(g) 2 ln
4 + x^2 + x
x
2
4 + x^2 + C
(h) −
x^2 + 2x + 2
x + 1
+ C
