Let $f:\mathbb R\to \mathbb R $ be a function whose constraint $f[0,1] $ is a continuous function. check if the integral exists and if so calculate it.
$$\lim_{n\rightarrow\infty}\int_{0}^{1}f(\frac{x^n}{1+x^2})dx$$.
How can I prove that it exists, just indicate that it is a continuous function in the constraint? Another questio, to calculate the integral, do I integrate by part ( integratte $x^n$ e derivative $1/1+x^2)$? Can you help me please?
For each $n\in \mathbb{N}$ define $f_n\colon [0,1]\to \mathbb{R}$, $f_n(x)=f(\frac{x^n}{1+x^2})$. Then, by the continuity of $f$ the sequence $(f_n)$ converges pointwise to the function $g:[0,1]\to \mathbb{R}$, $g(x)=f(0)$ for $x\in [0,1)$ and $g(1)=f(1/2)$. Now, you need some theorem, that allows you to interchange the limit and the integral:
$\lim_{n\to \infty} \int_{0}^1 f_n(x)dx\overset{?}=\int_{0}^1 \lim_{n\to \infty}f_n(x)dx=\int_{0}^1 g(x)dx=f(0)$.
The step with the question mark needs justification. Do you know a theorem that justifies this step?