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˜ DE INTEGRAIS LISTA DE EXERC´ICIOS- REVISAO

Esta lista ´e uma revis˜ao das t´ecnicas de integra¸c˜ao. Tais t´ecnicas ser˜ao necess´arias peara a resolu¸c˜ao dos problemas durante o curso de C´alculo III. Todos os exerc´ıcios s˜ao dos Cap´ıtulos 5 e 7 do J. Stewart, C´ alculo, Volume 1, 5a Edi¸c˜ao.

Se¸c˜ ao 5.5- Regra da Substitui¸c˜ ao, p´ ag. 412: 7, 11, 19, 21, 31, 53 e 59. Calcule a integral indefinida: 7-

Z

2x(x2 + 3)2 dx

11-

Z

1 + 4x √ dx 1 + x + 2x2

19-

Z

sin(πt) dt

21-

Z

(ln x)2 dx x

31-

Z

1 dx x ln x

√ 1 1 Respostas: 7. f (x) = (x2 + 3)5 + C, 11. f (x) = 2 1 + x + 2x2 + C, 19. f (x) = − cos π t + C, 5 π 1 21. f (x) = (ln x)3 + C, 31. f (x) = ln | ln x| + C . 3

Calcule a integral definida, se ela existir: 53-

Z

π

Z

π 3

sec2 (t/4) dt

0

59-

sen θ dθ cos2 θ

0

Respostas: 53. 4; 59. 1.

Se¸c˜ ao 7.1- Integra¸c˜ ao por Partes, p´ ag. 476: 5, 9, 13, 27, 33 e 35. Avalie a Integral: 5-

Z

r

re 2 dr

2

9-

Z

ln(2x + 1) dx

13-

Z

(ln x)2 dx

27-

Z

cos x ln(sin x) dx

1 (2x + 1) ln(2x + 1) − x + C , 2 2 13. f (x) = x(ln x) − 2x ln +2x + C, 27. f (x) = sin x(ln(sin x) − 1) + C . r

Respostas: 5. f (r) = 2(r − 2)e 2 + C, 9. f (x) =

Primeiro fa¸ ca uma substitui¸ c˜ ao e ent˜ ao use a integra¸ c˜ ao por partes para avaliar a integral. 33-

Z Z

√ x dx √ π

35- √ θ 3 cos(θ 2 ) dθ π 2

√ √ √ 1 π Respostas: 33.f (x) = 2(sin x − x cos x) + C, 35. − − . 2 4

Se¸c˜ ao 7.2- Integrais Trigonom´ etricas, p´ ag. 484: 3, 7, 9, 15, 17, 19, 23, 41 e 43. 379-

Z Z Z

3π 4

sin5 x cos3 x dx

π 2 π 2

cos2 θ dθ 0 π

sin4 (3t) dt 0

15-

Z

√ sin3 x cos x dx

17-

Z

cos2 x tan3 x dx

19-

Z

1 − sin x dx cos x

23-

Z

tan2 x dx

41-

Z

sin 5x sin 2x dx

43-

Z

cos 7θ cos 5θ dθ

3   √ 11 2 2 π π 3 Respostas: 3. f (x) = − cos x − cos x , 7. f (x) = , 9. f (x) = , 15. f (x) = cos x + C , 4 7 4 384 3 1 17. f (x) = cos2 x − ln | cos x| + C, 19. f (x) = ln(1 + sin x) + C , 23. f (x) = tan x − x + C, 2 1 1 1 1 sin 7x + C, 43. f (x) = sin 2θ + sin 12θ + C . 41. f (x) = sin 3x − 4 14 6 24

Se¸c˜ ao 7.3- Substitui¸c˜ ao Trigonom´ etrica, p´ ag. 490: 5, 7, 9, 17, 19, 23 e 27. Avalie a Integral: 5-

Z

7-

Z

9-

Z

2 √ 3 2t

1 √ dt t2 − 1

1 √ dx x2 25 − x2 1 √ dx x2 + 16

x √ dx 2 x −7 Z √ 1 + x2 19dx x Z p 235 + 4x − x2 dx

17-

Z

27-

Z

(x2

1 dx + 2x + 2)2

r √ √ π 25 − x2 3 1 Respostas: 5. f (x) = − , 7. f (x) = − + C , 9. f (x) = ln( x2 + 16 + x) + C , + 8 4 25x 24  √    p  ( 1 + x2 − 1)  p 2 17. f (x) = x − 7 + C, 19. f (x) = ln   + 1 + x2 + C ,   x     p 9 1 x−2 1 tan−1 (x + 1) + (x + 1) −1 2 23. f (x) = sin + (x − 2) 5 + 4x − x + C, 27. f (x) = + C. 2 2 2 3 (x2 + 2x + 2)

Se¸c˜ ao 7.4- Fra¸c˜ oes Parciais, p´ ag. 500: 7, 8, 9, 15, 21, 22, 23, 25, 27, 38, 39 e 43. Avalie a Integral: x dx x−6

7-

Z

8-

Z

r2 dr r+4

9-

Z

x−9 dx (x + 5)(x − 2)

4

1

2x + 3 dx (x + 1)2

15-

Z

21-

Z

5x2 + 3x − 2 dx x3 + 2x2

22-

Z

s2 (s

23-

Z

x2 dx (x + 1)3

25-

Z

10 dx (x − 1)(x2 + 9)

27-

Z

x3 + x2 + 2x + 1 dx (x2 + 1)(x2 + 2)

38-

Z

x4 + 1 dx x(x2 + 1)2

0

1 ds − 1)2

r2 Respostas: 7. x + 6 ln |x − 6| + C, 8. − 4r + 16 ln |r + 4| + C, 9. 2 ln |x + 5| − ln |x − 2| + C , 2 1 1 1 1 + C, 15. 2 ln 2 + , 21. 2 ln |x| + 3 ln |x + 2| + + C, 22. 2 ln |s| − 2 ln |s − 1| − − s (s − 1) x 2  x 2 1 1 1 23. ln |x + 1| + + C, 25. ln |x − 1| − ln(x2 + 9) − tan−1 − + C, 2 3 2 x +1 [2(x + 1) 3 ]  1 1 x 1 tan−1 √ + C, 38. ln |x| + 2 27. ln(x2 + 1) + √ + C. 2 (x + 1) 2 2

Fa¸ca a substitui¸ c˜ ao para expressar o integrando como uma fun¸ c˜ ao racional e ent˜ ao avalie a integral: 39-

1 √ dx x x+1

Z

x3 √ dx 3 x2 + 1 √   x+1−1  + C , 43. 3 (x2 + 1) 35 − 3 (x2 + 1) 23 + C. Respostas: 39. ln  √ 10 4 x+1+1 43-

Z

Se¸c˜ ao de Revis˜ ao- Exerc´ıcios, p´ ag. 536: 1, 7, 9, 13, 15, 21, 29, 31 e 37. Avalie a Integral: 17-

Z Z

0 5

x dx x + 10

sin(ln t) dt t

5

9-

Z

4 3

x 2 ln x dx 1

dx dx +x

13-

Z

15-

Z

sin2 θ cos5 θ dθ

21-

Z

dx √ dx x2 − 4x

Z

1

29-

ln 10

31-

Z

37-

Z

x3

x5 sec x dx

−1

0

√ ex ex − 1 dx ex + 8

(cos x + sin x)2 cos 2x dx

2 1 124 64 Respostas: 1. 5 + 10 ln , 7. − cos(ln t) + C , 9. , 13. ln |x| − ln(x2 + 1) + C , ln 4 − 2 25 5 3 √ 1 1 2 3π 15. sin3 θ − sin5 θ + sin7 θ + C, 21. ln |x − 2 + x2 − 4x| + C, 29. 0, 31. 6 − , 5 3 2 7 1 1 37. sin 2x − cos 4x + C . 2 8...