IMC 1995 · Problem 4

Let $F : (1, \infty) \to \mathbb{R}$ be the function defined by

$$F(x) := \int_x^{x^2} \frac{dt}{\ln t}.$$

Show that $F$ is one-to-one (i.e. injective) and find the range (i.e. set of values) of $F$.