Show that the series ∑_(n=1)^∞▒sin⁡nx/n^2 uniformly converges on R. verify that ∫_0^x▒〖( ∑_(n=1)^∞▒sin⁡nx/n^?

Observe that sin (nx) / n^2 is less than or equal to 1 / n^2 for all values of n and x. By the integral test, the series 1/n^2 converges.