Activity 6.5.3

Determine whether each of the following improper integrals converges or diverges. For each integral that converges, find its exact value.

1. $\int_1^{\infty} \frac{1}{x^2} \, dx$

2. $\int_0^{\infty} e^{-x/4} \, dx$

3. $\int_2^{\infty} \frac{9}{(x+5)^{2/3}} \, dx$

4. $\int_4^{\infty} \frac{3}{(x+2)^{5/4}} \, dx$

5. $\int_0^{\infty} x e^{-x/4} \, dx$

6. $\int_1^{\infty} \frac{1}{x^p} \, dx$, where $p$ is a positive real number