What are the ways to bound function for using the theorem of dominated convergence? I need to bound $f(x,n)=\frac{\sqrt{n}}{1+n\ln(1+x^2)}$, on $(0,1)$. I tried to use the fact that $a^2+b^2\geqslant 2ab$, but integral $\int\limits_0^1 \frac{1}{\sqrt{\ln(1+ x^2)}}\ dx$ is not convergent so I can't use theorem of dominated convergence.
Also I know about analysis methods if a function is convex, but I don't know how to use it.  I know about Jensen's inequality, but I need to know how with this to see if a function is bigger than some function that is integrable.

You do not need to bound it.
$f(x,n)$ is decreasing sequence of functions and $\int_0^1f(x,1)dx <+\infty$
Let $g(x,n)=f(x,1)-f(x,n)$
$f(x,n) \to 0 ,\forall x \in (0,1)\Longrightarrow g(x,n) \to f_1(x,n), \forall x \in (0,1)$
Thus $\int_0^1g(x,n)dx \to \int_0^1 f(x,1)dx \Longrightarrow \int_0^1 f(x,n)dx \to 0$
By monotone convergence $
