Theorem: (Dominated Convergence Theorem). Let $(f_n)_{n\geq 1}$ be a sequence of integrable functions such that $\lim\limits_{n\to\infty}f_n = f$ pointwise. If there exists an integrable function $g:S\to[0,\infty]$ such that $\left|f_n\right|\leq g$ for all $n\geq 1$, then $f$ is integrable and $\lim\limits_{n\to\infty}\int_Sf_n d\mu = \int_S f d\mu$.
I have to use this theorem to solve the following exercise:
Exercise: Find $\lim\limits_{n\to\infty}\displaystyle\int_{\mathbb{R}}\mathbb{1}_{[0,1]}(x)\dfrac{nx^n\sin(nx)-n}{\sqrt{x + 2n^2}}d\lambda$, using DCT (answer is $-\frac{1}{2}\sqrt{2}$).
What I've tried: I have to find $\lim\limits_{n\to\infty}f_n = f$ and a function $g$ such that $\left|f_n\right|\leq g$ for all $n\geq 1$. This would imply that $f$ is integrable so that I can proceed to calculate the integral according to the DCT. Unfortunately I'm not able to determine $\lim\limits_{n\to\infty}f_n$. I have that $$\lim\limits_{n\to\infty}\mathbb{1}_{[0,1]}(x)\dfrac{nx^n\sin(nx)-n}{\sqrt{x + 2n^2}} = \lim\limits_{n\to\infty}\mathbb{1}_{[0,1]}(x)\dfrac{x^n\sin(nx)-1}{\sqrt{x + 2}}.$$ $-2\leq x^n\sin(nx)-1 \leq 0$, which means that the limit is oscillating. I'm not sure how to proceed from here.
Question: How do I solve this exercise using the given theorem?
Thanks in advance!