12. Techniques of Integration

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.

36. ∫ (from e² to ∞) 1/(x lnᵖ x) dx, p > 1

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to 1 of (dt / (t - sin t))

(Hint: t ≥ sin t for t ≥ 0)

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

22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx

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).

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀^∞ dx / [(x + 1)(x² + 1)]

88. Incorrect Calculation

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

90. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / √x, y = 0, x = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (i) about the x-axis

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋∞² (2 dx) / (x² + 4)

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋₈¹ dx / x^(1/3)

77–86. Comparison Test Determine whether the following integrals converge or diverge.

81. ∫(from 1 to ∞) (sin²x) / x² dx

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (4r dr) / √(1 − r⁴)

62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.

a. Find the probability that the computer chip fails after 15,000 hr of operation.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of (dx / √(x² - 1))

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)