List of top Real Analysis Questions on Sequences of Functions and Limit of Integrals

Let \[ \alpha = \lim_{n \to \infty} \int_0^{1} \frac{n^2 + (\sin e^x)^n}{7n^2 + x^8} \, dx. \] The value of \(14\alpha\) is equal to ______. (Answer in integer)