Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$

Question

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$

Attempt: We define $f_n(x)=\frac{n+\cos^n(e^x)}{4n+x^4}$ on the domain $(0,1)$.

This sequence of functions $(f_n)$ converges pointwise to $\frac{1}{4}$. This is dominated by $g(x)=\frac{1}{2}$.

The set $[0,1]$ is measureable and continuous functions are measureable as well.

Now, using Lebesgue dominated convergence theorem, we can interchange the limit and we finally get $\int_{0}^{1}\frac{1}{4}dx=\frac{1}{4}$.

Where did I go wrong?