Convergence of an integral where integrand oscillates faster than sawtooth

Question

While studying convergence from Arfken and Weber (Mathematical Methods for Physicicts), I encountered following integral :

$$g(N)=\int^N_1{(n-[n])sin(nx)dn}$$

How can I show that $g(n)$ is bounded above? I can see that when $n$ is large, sin function oscillates faster than the sawtooth so within its period the function would take some negative values as well which might annihilate positive values and thus may bound it. But I don't know how do I show it explicitly.

Answer 1

In closed form, $$g(N)=\left(\frac{\sin\frac{Nx}2\cos\frac{(N-1)x}2}{\sin\frac x2}-1\right)\frac{\sin x-x\cos x}{x^2}+\frac{\sin\frac{Nx}2\sin\frac{(N-1)x}2}{\sin\frac x2}\frac{\cos x+x\sin x-1}{x^2}$$ ($N\in\mathbb N$)

Now you can analyse.