12. Techniques of Integration

Which of the following integrals is improper?

Which of the following integrals is improper because the interval of integration is infinite?

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