12. Techniques of Integration

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 (ii) about the y-axis.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₋∞⁰ x² e^(x³) dx

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 ∞ of (dθ / (θ² - 1))

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

∫₂⁴ dt / [t√(t² − 4)]

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

42. ∫ (from 3 to 4) 1/(x-3)³ᐟ² dx

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

47. ∫ (from 0 to 10) 1/∜(10 - x) dx

92. Evaluate ∫ from 3 to ∞ [ dx / (x √(x² - 9))]

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)?

Evaluate the integral or state that it diverges. 

${\int }_2^{\infty }\frac{1}{x(lnx{)}^4}dx$

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 1 of (-x ln|x| dx)

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 2 of (dx / (x ln x))

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

78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

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

∫₁^∞ dx / [x√(x² − 1)]

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₂^∞ (1 / (x√x)) dx

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.