and I am trying to show that the integral $\int_{\mathbb{R}}\sin(nx)fdx$ converges to $0$ for any test function $f$ while $\int_{\mathbb{R}}\sin^2(nx)f(x)dx$ does not go to 0.
For the first function, the symmetry of the $\sin(nx)$ is what gets you the "0 integral" I presume. As $n\to \infty$ the integral on any region is 0 for a constant test function but apart from this. I am not sure what to do.
For the second one, choose a constant test function on it's support. Since $\sin(nx)^2$ is always non-negative, we can find some sort of lower bound for the integral so that we don't have convergence.