EDIT: The first comment on this post brought to my attention the error in my logic. I have since figured out where I went wrong!
We have a sequence of functions on the domain $x\in [0,1]$ which looks like this: $f_n(x)=(1+x^2+\frac{x^4}{n})^{-1}$. From my understanding, if $f_n$ is Riemann integrable on our domain and the sequence converges uniformly then $\lim_{n\to\infty}\int_0^1 f_n(x)dx=\int_0^1f(x)dx$ where $f(x)=\lim_{n\to\infty}f_n(x)$. I then broke it up into cases to check for uniform convergence and got that for $x=0$ we see that the sequence of functions converges to $f_n(x) \to 1$. Then through a similar process, I checked that for $x=1$ that $\lim_{n\to\infty}f_n(x)=\frac{1}{2}$. Then to check anything in between I checked $x=\frac{1}{n}$ and got that $\lim_{n\to\infty} f_n(x)=1$. Did I do something wrong? Because this looks like our limit function is discontinuous so I'm not sure how I would go about integrating it? I feel like I must have done something wrong so any guidance on how to complete this integral would be great! Thank you :)
Minor edit: Please feel free to correct my formatting! I am very new to using math stack so I want to make sure I learn how to use this resource properly