Determine whether each of the following integrals is proper, improper and convergent, or improper and divergent.

Question

Determine whether each of the following integrals is proper, improper and convergent, or improper and divergent. (I keep getting one wrong, not sure which).

1. ∫[from 2 to ∞] (1 / √(x^3 + 8)) dx (I put Improper and convergent)
2. ∫[from -∞ to 0] (x^4 * e^(x^5)) dx (I put Improper and convergent)
3. ∫[from 4 to 8] (1 / (x - 5)^3) dx (I put Proper)
4. ∫[from 0 to ∞] (x^5 * e^(x^6)) dx (I put Improper and divergent)
5. ∫[from 5 to ∞] (1 / ∛(x^2 - 4)) dx (I put Improper and divergent)
6. ∫[from -∞ to ∞] (x / (x^2 + 12)) dx (I put Improper and divergent)
7. ∫[from -∞ to ∞] cos(4x) dx (I put Improper and divergent)
8. ∫[from 0 to ∞] (arctan(x) / (1 + x^2)) dx (I put Improper and convergent)

Answer 1

∫_{(4 to 8)} [1 / (x-5)³]dx is improper since the integrand is undefined when x = 5.