Exercícios complementares_Unidade 5

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UNIVERSIDADE FEDERAL DE SANTA CATARINA 
CAMPUS DE JOINVILLE 
CURSO DE ENGENHARIA DA MOBILIDADE 
EMB 5001 – CÁLCULO DIFERENCIAL E INTEGRAL I 
 
Professores: Alexandre Mikowski e Rafael Machado Casali 
 
EXERCÍCIOS COMPLEMENTARES 
 
Unidade 5 – Integrais definidas e indefinidas 
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1 – Encontrar uma primitiva F, da função xxxf += 32)( , que satisfaça .1)1( =F 
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2 – Determinar a função )(xf tal que .2cos
2
1)( 2 cxxdxxf ++=∫ 
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3 – Encontrar uma primitiva da função 1
1)( 2 += xxf que se anule no ponto x = 2. 
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4 – Calcular as integrais indefinidas. 
a) ∫ 3x
dx
 
b) ∫ 






+ dt
t
t
3
2 19
 
c) ( )∫ ++ dxcbxax 334 
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d) ∫ 






+ dxxx
x 3
1
 
e) ( )∫ − dxx 32 2 
f) ∫ 






− dy
y
y
2
12
 
g) dxxx∫
3
 
h) ∫
−+ dx
x
xx
4
25 12
 
i) ∫ dxx
x
2cos
sen 
 
j) ∫
−
dx
x21
9
 
k) ∫
−
dx
xx
24
4
 
l) dx
x
xxxx
∫
+−+−
2
234 12698
 
m) dt
t
t
et
∫ 





++
1
2
 
n) θθθ d tgcos∫ ⋅ 
o) ( )dxee xx∫ −− 
p) ( )dtttttt∫ ++++ 543 
q) dx
x
x
∫
−
− 531
 
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r) ( )dttett∫ +− cosh22 
s) dt
t
tet∫ 





+− 3
4 316
 
t) ( ) ( ) dxxx 11 22 +−∫ . 
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5 – Calcular as integrais definidas. 
[1] ( )∫
−
+
2
1
31 dxxx
 
[2] ( )∫
−
+−
0
3
2 74 dxxx
 
[3] ∫
2
1
6x
dx
 
[4] dttt∫
9
4
2
 
[5] ∫ +
1
0 13y
dy
 
[6] dxx∫
pi2
0
sen 
 
[7] dtt∫
−
−
5
2
42
 
[8] dxxx∫ +−
4
0
2 23
 
[9] ( )∫ +
4
1
3
1xx
dx
 
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[10] ( ) dxx 12
4
0
21
∫
−+
 
[11] ( )dxxx∫ +2
0
52
 
[12] dx
x
xxx
∫
+−+2
1
2
23 2575
 
[13] dt
t
t∫
−
−






−
2
3
2
 
1
. 
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6 – Encontrar a área da região limitada pelas curvas dadas. 
a) 2 e ,21 +−=== xyyxx 
b) yxxy 2 e 2 22 == 
c) 3 e 5 2 +=−= xyxy 
d) 6 e 
6
1 2
== yxy
 
e) 3 e 1 2 −=−= yxy 
f) 3 e 3 2 =+=+ xyyx 
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7 – Calcular as integrais usando o método da substituição. Derivar o resultado da integral obtida e verificar se 
encontra a f (x). 
[1] ( ) ( )∫ +−+ dxxxx 12322 102 
[2] ( )∫ − dxxx 2713 2 
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[3] ∫
−
5 2 1x
xdx
 
[4] ∫ − dxxx
2345
 
[5] ∫ + dxxx
42 2
 
[6] ( )∫ + dtee tt 2312 2 
[7] ∫ + 4t
t
e
dte
 
[8] ∫
+ dx
x
e x
2
1
2
 
[9] ∫ dxxx sec tg
2
 
[10] ∫ dxxx cos sen
4
 
[11] ∫ dxx
x
5cos
sen 
 
[12] ∫
− dx
x
xx
 cos
 cos 5sen 2
 
[13] ∫ dxee
xx
 2 cos
 
[14] ∫ dxx
x
 2
 cos 
2
 
[15] ( )∫ − θpiθ d 5sen 
[16] ∫
−
dy
y
y
 
12
sen arc
2
 
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[17] ∫ +
θ
θ
θ d
ba
 
 tg
sec 2 2
 
[18] ∫ +
 
16 2x
dx
 
[19] ∫ +−
 
442 yy
dy
 
[20] θθθ d cos sen3∫ 
[21] dx
x
x
∫
2ln 
 
[22] ( ) dxee axax∫ −+ 2 
[23] dttt∫ +
243
 
[24] ∫ ++ 34204
4
2 xx
dx
 
[25] ∫ +− 14
3
2 xx
dx
 
[26] ∫ +162x
x
e
dxe
 
[27] ∫
−
+ dx
x
x
1
3
 
[28] ∫ xx
dx
3ln 
3
2 
[29] ( )∫ + dxx 2 cos4sen pi 
[30] ∫
+ dxxx 2 1
2
 
[31] ∫ dxxe
x
23
 
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[32] ( )∫ + 22 t
dt
 
[33] ∫ tt
dt
ln 
 
[34] ∫ − dxxx
2218
 
[35] ( )∫ + dxee xx 252 2 
[36] ( )∫ + 51 vv
dv
 
[37] ∫
+ 54
4
2t
tdt
 
[38] ∫ dxx
x
sen -3
 cos
 
[39] ∫ + dxxx 1
2
 
[40] ∫
− dxex x 
54
 
[41] ∫ dttt cos 
2
 
[42] ∫ + dxxx 568
32
 
[43] ∫ θθθ d 2 cos 2sen 2
1
 
[44] ( )∫ + dxx 35sec2 
[45] ( )∫ − 3cos5
 sen
θ
θ dθ
 
[46] u du cotg 
[47] ( ) 0 ,1 23 >+∫ −− adtee atat 
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[48] dx
x
x
∫
cos
 
[49] dttt 4∫ − 
[50] ( )( )dxxxx∫ + 42sen 32 
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8 – Resolver as seguintes usando a técnica de integração por partes. Derivar o resultado da integral obtida e 
verificar se encontra a f (x). 
1) ∫ dxxx 5sen 
2) ( )∫ − dxx 1ln 
3) ∫ dtet
t4
 
 
4) ( )∫ + dxxx 2 cos 1 
5) ∫ dxxx 3ln 
6) ∫ dxx cos
3
 
7) ∫ dx
x
e x 
2
cos 
 
8) ∫ dxxx ln 
9) ∫ dxx cosec
3 ] 
10) ∫ dxaxx cos 
2
 
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Observações: 
I. Os exercícios de 1 até 6 e 8 até 18 propostos foram selecionados do livro: FLEMING, D. M. & 
GONÇALVES, M. B. Cálculo A. Vol. 1; 6ª edição, Pearson Prentice Hall, São Paulo, 2007. Capítulo 
6. 
II. A lista de exercícios corresponde aos tópicos da ementa: Integral definida e indefinida 
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