12. Techniques of Integration

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

∫₋∞⁴ [x / (x² + 9)^(2/5)] dx

Evaluate the improper integrals in Exercises 53–62.

∫ from 3 to ∞ of (2 / (u² − 2u)) du

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

19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx

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

∫₁^∞ (1 / (x² + 3x)) dx

94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.

For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?

What is its value when it exists?

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)