Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?
- $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$
- $\int_{1}^{\infty} \frac{f(x)}{1+x} \mathrm{~d} x=\infty$
- $\int_{1}^{\infty}(\ln x) f(x) \mathrm{d} x=\infty$
- $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$
- None of the above