12. Techniques of Integration

Which of the following integrals is improper because the interval of integration is infinite?

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

85. Find the volume of the solid generated by revolving the region about the y-axis.

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 4 to ∞ of (dx / (√x - 1))

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

39. ∫ (from 0 to π/2) tan θ dθ

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ

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

10. ∫ (from 0 to ∞) e⁻²ˣ dx

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

y = 1 / x², y = 0, and x = 1.

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

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

31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv

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 ((1 / ln x) dx)

63. Average Lifetime The average time until a computer chip fails (see Exercise 62) is 0.00005 ∫(from 0 to ∞) t e^(-0.00005t) dt. Find this value.

Evaluate the integral or state that it diverges. 

${\int }_{-\infty }^{-1}\frac{2}{x^3}dx$

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

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

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

56. ∫ (from 0 to 1) 1/(x + √x) dx

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

∫₁^∞ dx / x^1.001

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

16. ∫ (from -∞ to ∞) (1/(x² + a²)) dx, a > 0