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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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

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

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

∫₁^∞ (1 / (x² + 3x)) dx

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

∫₂^∞ (1 / (x√x)) dx

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

83. Find the area of the region.

For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.

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

∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ