12.6.3 Higher mathematics

Problem: Find all functions, differentiable on \( \mathbb{R} \), \( f: \mathbb{R} \rightarrow[1 ;+\infty) \) for which \[ \int_{1}^{f(x)} e^{u^{2}} d u=\int_{0}^{x} \frac{u d u}{f(u)}, \quad \forall x \in \mathbb{R} . \]