68 ολοκληρώματα για την ημέρα 7/4/2025

Να υπολογιατούν τα ολοκληρώματα: 

1. ${\displaystyle \int x{\sec }^{2}xdx}$ 

2. ${\displaystyle \int \frac{\cos \sqrt{x}}{\sqrt{x}}dx}$ 

3. ${\displaystyle \int x(\ln x{)}^{2}dx}$ 

4. ${\displaystyle \int \frac{1}{x(\ln x{)}^2}dx}$ 

5. ${\displaystyle \int \frac{\ln x}{x^2}dx}$ 

6. ${\displaystyle \int \frac{(\ln x{)}^3}{x}dx}$ 

7. ${\displaystyle \int x^3{e}^{x}dx}$ 

8. ${\displaystyle \int x^5{e}^{x}dx}$ 

9. ${\displaystyle \int x^3\sqrt{x^2+1}dx}$ 

10. ${\displaystyle \int x^2\sin x^{3}dx}$ 

11. ${\displaystyle \int \sin 3x\cos 2xdx}$ 

12. ${\displaystyle \int \sin 2x\cos 4xdx}$ 

13. ${\displaystyle \int \sqrt{x}\ln xdx}$ 

14. ${\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx}$ 

15. ${\displaystyle \int \cos \sqrt{x}dx}$ 

16. ${\displaystyle \int \sqrt{x}{e}^{\sqrt{x}}dx}$

17. ${\displaystyle {\int }_0^{\pi /2}{\theta }^2\sin 2\theta d\theta }$

18. ${\displaystyle {\int }_0^{\pi /2}{x}^3\cos 2xdx}$ 

19. ${\displaystyle {\int }_{2/\sqrt{3}}^{2}t{\sec }^{-1}tdt}$ 

20. ${\displaystyle {\int }_0^{1/\sqrt{2}}2x{\sin }^{-1}(x^2)dx}$ 

21. ${\displaystyle \int x{\tan }^{-1}xdx}$ 

22. ${\displaystyle \int x^2{\tan }^{-1}\frac{x}{2}dx}$ 

23. ${\displaystyle \int \frac{dx}{\sqrt{9+x^2}}}$ 

24. ${\displaystyle \int \frac{3dx}{\sqrt{1+9x^2}}}$ 

25. ${\displaystyle {\int }_{-2}^2\frac{dx}{4+x^2}}$ 

26. ${\displaystyle {\int }_0^2\frac{dx}{8+2x^2}}$ 

27. ${\displaystyle {\int }_0^{3/2}\frac{dx}{\sqrt{9-x^2}}}$ 

28. ${\displaystyle {\int }_0^{1/2\sqrt{2}}\frac{2dx}{\sqrt{1-4x^2}}}$ 

29. ${\displaystyle \int \sqrt{25-t^2}dt}$ 

30. ${\displaystyle \int \frac{dx}{\sqrt{4x^2-49}},x>\frac{7}{2}}$ 

31. ${\displaystyle \int \frac{5dx}{\sqrt{25x^2-9}},x>\frac{3}{5}}$ 

32. ${\displaystyle \int \frac{\sqrt{y^2-49}}{y}dy,y>7}$ 

33. ${\displaystyle \int \frac{\sqrt{y^2-25}}{y^3}dy,y>5}$ 

34. ${\displaystyle \int \frac{dx}{x^2\sqrt{x^2-1}},x>1}$ 

35. ${\displaystyle \int \frac{2dx}{x^3\sqrt{x^2-1}},x>1}$

36. ${\displaystyle \int \sqrt{1-9t^2}dt}$ 

37. ${\displaystyle \int \frac{dx}{x\sqrt{x-3}}}$ 

38. ${\displaystyle \int \frac{dx}{x\sqrt{x+4}}}$  

39. ${\displaystyle \int \frac{xdx}{\sqrt{x-2}}}$ 

40. ${\displaystyle \int \frac{xdx}{(2x+3{)}^{3/2}}}$ 

41. ${\displaystyle \int \sqrt{2x-3}dx}$ 

42. ${\displaystyle \int \frac{\sqrt{9-4x}}{x^2}dx}$ 

43. ${\displaystyle \int \frac{x^2}{\sqrt{4x-9}}dx}$ 

44. ${\displaystyle \int \sqrt{4x-x^2}dx}$ 

45. ${\displaystyle \int \sqrt{\frac{x-x^2}{x}}dx}$ 

46. ${\displaystyle \int \frac{dx}{\sqrt{7+x^2}}}$ 

47. ${\displaystyle \int \frac{dx}{\sqrt{7-x^2}}}$ 

48. ${\displaystyle \int \frac{\sqrt{4-x^2}}{x}dx}$ 

49. ${\displaystyle \int \sqrt{x^2-4}dx}$ 

50. ${\displaystyle \int e^{2t}\cos 3tdt}$ 

51. ${\displaystyle \int e^{-3t}\sin 4tdt}$ 

52. ${\displaystyle \int x{\cos }^{-1}xdx}$ 

53. ${\displaystyle \int x{\tan }^{-1}xdx}$ 

54. ${\displaystyle \int x^2{\tan }^{-1}xdx}$ 

55. ${\displaystyle \int \frac{{\tan }^{-1}x}{x^2}dx}$ 

56. ${\displaystyle \int \sin 3x\cos 2xdx}$ 

57. ${\displaystyle \int \sin 2x\cos 3xdx}$ 

58. ${\displaystyle \int 8\sin 4t\sin \frac{t}{2}dt}$ 

59. ${\displaystyle \int \sin \frac{t}{3}\sin \frac{t}{6}dt}$ 

60. ${\displaystyle \int \cos \frac{\theta }{3}\cos \frac{\theta }{4}d\theta }$ 

61. ${\displaystyle \int \cos \frac{\theta }{2}\cos 7\theta d\theta }$

62. ${\displaystyle \int (7x+5{)}^{3/2}dx}$ 

63. ${\displaystyle {\int }_0^{\pi /2}\tan \theta d\theta }$ 

64. ${\displaystyle {\int }_0^{\pi /2}\cot \theta d\theta }$ 

65. ${\displaystyle {\int }_0^1\frac{\ln x}{x^2}dx}$ 

66. ${\displaystyle {\int }_1^2\frac{dx}{x\ln x}}$ 

67. ${\displaystyle {\int }_0^{\ln 2}{x}^{-2}{e}^{-1/x}dx}$ 

68. ${\displaystyle {\int }_0^1\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx}$