The sequence $\{x_{n}\}$ by $x_{n}=\frac{1}{2\pi}\int_{0}^{\pi/2}\tan^{\frac{1}{n}}t \ dt$ is such that $\{x_{n}\}$ converges to $1/4$ For $n\geq 2$, define the sequence $\{x_{n}\}$ by $$x_{n}=\frac{1}{2\pi}\int_{0}^{\pi/2}\tan^{\frac{1}{n}}t \ dt$$ Then prove that the sequence $\{x_{n}\}$ converges to $1/4$.
My Attampt: I think it is a application of Dominated convergence theorem, But I am not sure, give some Hints.

Divide the integral into two parts, from $0 < t < {\pi \over 4}$ and ${\pi \over 4} \leq t < {\pi \over 2}$. On the first interval one has $\tan t <  1$ and on the second one has $\tan t \geq 1$. Hence the functions $\tan^{1 \over n} t$ are increasing in $n$ on the left interval and decreasing in $n$ on the right interval. This should make it easier to find the right convergence theorem(s) to use on a given interval.
