A second set of 50 questions on Improper Integrals.
1. The integral \( \int_{1}^{\infty} \frac{dx}{x^{1.01}} \) is:
2. The integral \( \int_{0}^{1} \frac{dx}{x^{0.99}} \) is:
3. Test the convergence of \( \int_{1}^{\infty} \frac{e^{-x}}{x} \, dx \).
4. What is the value of \( \int_{-\infty}^{0} e^{2x} \, dx \)?
5. The integral \( \int_{0}^{4} \frac{dx}{x-2} \) is:
6. Using the Limit Comparison Test, investigate \( \int_{1}^{\infty} \frac{1}{x + \sqrt{x}} \, dx \).
7. The integral \( \int_{0}^{\infty} \frac{\arctan x}{1+x^2} \, dx \) is:
8. What is the type of integral \( \int_{0}^{1} \frac{dx}{x^2 - 4} \)?
9. Does \( \int_{1}^{\infty} \frac{1-e^{-x}}{x} \, dx \) converge or diverge?
10. The integral \( \int_{0}^{1} \frac{dx}{\sqrt[3]{1-x}} \) is:
11. Test the convergence of \( \int_{2}^{\infty} \frac{dx}{x^2 - 1} \).
12. The integral \( \int_{0}^{\infty} \frac{x^3}{e^x} \, dx \) is related to which function?
13. Does \( \int_{0}^{\pi/2} \sec(x) \, dx \) converge or diverge?
14. The integral \( \int_{1}^{\infty} \frac{\cos^2 x}{x^3} \, dx \) is:
15. What is the value of \( \int_{0}^{1} \frac{dx}{\sqrt{1-x}} \)?
16. The integral \( \int_{-\infty}^{\infty} \frac{x}{1+x^2} \, dx \) is:
17. Test the convergence of \( \int_{0}^{1} \frac{\ln x}{x} \, dx \).
18. The integral \( \int_{1}^{\infty} \frac{dx}{x \sqrt{x^2-1}} \) is:
19. Does \( \int_{0}^{\infty} \frac{\sin x}{x} \, dx \) (Dirichlet Integral) converge or diverge?
20. The integral \( \int_{0}^{1} \frac{1}{e^x - e^{-x}} \, dx \) is:
21. For what values of p does \( \int_{e}^{\infty} \frac{dx}{x(\ln x)^p} \) converge?
22. The integral \( \int_{0}^{\infty} e^{-x} \cos(x) \, dx \) is:
23. Evaluate \( \int_{1}^{\infty} \frac{dx}{x^3} \).
24. The integral \( \int_{0}^{1} \frac{dx}{x - x^2} \) is:
25. Test the convergence of \( \int_{1}^{\infty} \frac{x^2}{2^x} \, dx \).
26. The integral \( \int_{0}^{\infty} \frac{dx}{x^2 + 3x + 2} \) is:
27. Does \( \int_{0}^{1} \frac{\sin x}{\sqrt{x}} \, dx \) converge or diverge?
28. The integral \( \int_{1}^{\infty} \frac{dx}{e^x - 2^x} \) is:
29. What is the Cauchy Principal Value of \( \int_{-1}^{1} \frac{1}{x^3} \, dx \)?
30. The integral \( \int_{0}^{\infty} \frac{1}{x^2} \, dx \) is:
31. Test the convergence of \( \int_{1}^{\infty} \frac{\ln(x)}{x^2} \, dx \).
32. The integral \( \int_{0}^{1} \frac{dx}{x^2+x} \) is:
33. Does \( \int_{1}^{\infty} \frac{x^2-1}{x^4+1} \, dx \) converge or diverge?
34. The integral \( \int_{0}^{1} \frac{dx}{\sqrt{x-x^2}} \) is:
35. Test the convergence of \( \int_{1}^{\infty} \frac{1}{\sqrt{x^3+1}} \, dx \).
36. The integral \( \int_{0}^{\infty} x e^{-x^2} \, dx \) is:
37. Does \( \int_{0}^{1} \frac{dx}{\sin x} \) converge or diverge?
38. The integral \( \int_{1}^{\infty} \frac{1}{x+e^x} \, dx \) is:
39. Test the convergence of \( \int_{0}^{1} \frac{dx}{1 - \cos x} \).
40. The integral \( \int_{1}^{\infty} \frac{dx}{x^{p}} \, dx \) converges if and only if:
41. Does \( \int_{0}^{\infty} \frac{dx}{1+x^3} \) converge or diverge?
42. The integral \( \int_{0}^{1} \frac{dx}{\ln(x+1)} \) is:
43. Test the convergence of \( \int_{1}^{\infty} \frac{1}{x^2 + \sin x} \, dx \).
44. The integral \( \int_{0}^{1} \frac{dx}{x^p} \) converges if and only if:
45. Does \( \int_{1}^{\infty} \frac{1}{x(1+\ln x)} \, dx \) converge or diverge?
46. The integral \( \int_{0}^{\infty} \frac{e^{-x}}{\sqrt{x}} \, dx \) is:
47. Test the convergence of \( \int_{1}^{\infty} \frac{x^x}{e^x} \, dx \).
48. The integral \( \int_{0}^{1} \frac{dx}{\sqrt[3]{x^2}} \) is:
49. Does \( \int_{1}^{\infty} \frac{\sin(x)}{x^2} \, dx \) converge absolutely?
50. The integral \( \int_{1}^{\infty} \frac{dx}{x^{1/x}} \) is: