12. Techniques of Integration

Evaluate the integral or state that it diverges. 

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

Evaluate the integral or state that it diverges. 

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

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 

c. Find the minimum decay constant k for which the total oil reserves will last forever.

4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.

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

10. ∫ (from 0 to ∞) e⁻²ˣ dx

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

10. ∫ (from 0 to ∞) e⁻²ˣ dx

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

13. ∫ (from 0 to ∞) cos x dx

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

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

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

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

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

22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx

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

25. ∫ (from -∞ to ∞) e³ˣ/(1 + e⁶ˣ) dx

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

33. ∫ (from 2 to ∞) 1/(y ln y) dy

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

36. ∫ (from e² to ∞) 1/(x lnᵖ x) dx, p > 1

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

39. ∫ (from 0 to π/2) tan θ dθ

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

42. ∫ (from 3 to 4) 1/(x-3)³ᐟ² dx