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

81. ∫(from 1 to ∞) (sin²x) / x² dx

7–64. Integration review Evaluate the following integrals.

53. ∫ eˣ sec(eˣ + 1) dx

23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

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

13. ∫ (from 0 to ∞) cos x dx

23-64. Integration Evaluate the following integrals.

23. ∫ [3 / ((x - 1)(x + 2))] dx

23-64. Integration Evaluate the following integrals.

44. ∫₁² 2/[t³(t + 1)] dt

66-68. Areas of regions (Use of Tech) Find the area of the following regions.

66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1

7–84. Evaluate the following integrals.

82. ∫ 1/(1 + tanx) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

19. ∫ 1/√(x² - 81) dx, x > 9

59. Area of a segment of a circle

Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:

A_seg = (1/2) r² (θ - sin θ)

b. Find the area using calculus.

1. Give some examples of analytical methods for evaluating integrals.

41–48. Geometry problems Use a table of integrals to solve the following problems.

43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.

19-22. {Use of Tech} Trapezoid Rule approximations. Find the indicated Trapezoid Rule approximations to the following integrals.

21. ∫(0 to 1) sin(πx) dx using n = 6 subintervals

48. Integral of sec³x Use integration by parts to show that:

∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

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

9–40. Integration by parts Evaluate the following integrals using integration by parts.

11. ∫ t · e⁶ᵗ dt

15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.

18. ∫(0 to 1) e⁻ˣ dx using n = 8 subintervals

9–61. Trigonometric integrals Evaluate the following integrals.

51. ∫ (csc²x + csc⁴x) dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

20. ∫ sin⁻¹(x) dx

41–48. Geometry problems Use a table of integrals to solve the following problems.

46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

5. What is a reduction formula?

7–84. Evaluate the following integrals.

59. ∫ 1/(x⁴ + x²) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

24. ∫ dt / √(1 + 4eᵗ)

9–40. Integration by parts Evaluate the following integrals using integration by parts.

23. ∫ x² sin(2x) dx

Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.

7–84. Evaluate the following integrals.

9. ∫ from 4 to 6 [1 / √(8x – x²)] dx

Let f(x) = √(x + 1). Find the area of the surface generated when:

Region bounded by f(x) and the x-axis on [0, 1]

Revolved about the x-axis

72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].

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

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

"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by

Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),

where k is a physical constant and a > 0.

a. Confirm that Eₓ(a)=kQ / a √(a²+L²)

b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.