However the [tex]x_i[/tex] are chosen, the sum is equivalent to the definite integral,
[tex]\displaystyle\lim_{n\to\infty}\sum_{i=1}^n x_i\cos x_i\Delta x=\int_0^{2\pi}x\cos x\,\mathrm dx[/tex]
which can be computed via integration by parts. If [tex]u=x[/tex] and [tex]\mathrm dv=\cos x\,\mathrm dx[/tex], you have
[tex]\displaystyle\int_0^{2\pi}x\cos x\,\mathrm dx=[uv]_0^{2\pi}-\int_0^{2\pi}v\,\mathrm du[/tex]
or equivalently,
[tex]\displaystyle\int_0^{2\pi}x\cos x\,\mathrm dx=[x\sin x]_0^{2\pi}-\int_0^{2\pi}\sin x\,\mathrm dx=[\cos x]_0^{2\pi}=0[/tex]
