Exercise 11

Determine, with justification, whether each of the following improper integrals converges or diverges.

1. $\int_e^{\infty} \frac{\ln(x)}{x} \, dx$

2. $\int_e^{\infty} \frac{1}{x\ln(x)} \, dx$

3. $\int_e^{\infty} \frac{1}{x(\ln(x))^2} \, dx$

4. $\int_e^{\infty} \frac{1}{x(\ln(x))^p} \, dx$, where $p$ is a positive real number

5. $\int_0^1 \frac{\ln(x)}{x} \, dx$

6. $\int_0^1 \ln(x) \, dx$