What is the period of $x\sin x$?

Question

What is the period of $x\sin x$?

I’m not able to solve after $$(x+t)\sin(x+t)=x\sin x$$

Answer 1

This is because this function is not periodic. Suppose in fact that $(x + t) \sin(x + t) = x \sin(x)$ for $t > 0$. Then $$ \forall n \in \mathbb N: \frac{\pi}{2} = \frac{\pi}{2} \sin\left( \frac{\pi}{2} \right) = \left(nt + \frac{\pi}{2}\right) \sin\left( nt + \frac{\pi}{2} \right). $$ Yet either $t$ is a fractional multipl of $\pi$, leading to an immediate conradiction by preiodicity of $\sin$, or it is not, in which case the orbit of the action by $n$ is dense in the values o $\sin$ (as the corresponding dynamical system is minimal), leading to another contradiction.

Answer 2

Since $f(x) = x\sin x$ has $f((2n + \frac{1}{2})\pi) = (2n + \frac{1}{2})\pi$ for any integer $n$, it is unbounded and thus not periodic.