87. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

88. Incorrect Calculation

b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

82. ∫ (from -∞ to -1) dx/(x - 1)⁴

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*

89–91. Comparison Test Determine whether the following integrals converge or diverge.

89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

92. Integral with a parameter For what values of p does the integral

∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?

What are the two general ways in which an improper integral may occur?

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

28. ∫ (from 1 to ∞) tan⁻¹(s)/(s² + 1) ds