Let $u = x$ and $dv = \sin x\,dx$. Then, $du = dx$ and $v = -\cos x$.
Using integration by parts, we have:
$\int_0^\infty x\sin x\,dx = \left[-x\cos x\right]_0^\infty + \int_0^\infty \cos x\,dx$
Now, we need to evaluate the limit:
$\lim_{t\to\infty}
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