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. 

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

Which of the following integrals is improper?

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

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