Justify that, if $\varphi$ is a $\mathcal{C}^1$ application from $[0, \pi]$ to $\mathbf{R}$, then $$\lim_{x \to +\infty} \int_0^{\pi} \varphi(t) \sin(xt) \mathrm{d}t = 0$$ and conclude that $$\sigma(1) = \frac{\pi^2}{6}$$
Justify that, if $\varphi$ is a $\mathcal{C}^1$ application from $[0, \pi]$ to $\mathbf{R}$, then
$$\lim_{x \to +\infty} \int_0^{\pi} \varphi(t) \sin(xt) \mathrm{d}t = 0$$
and conclude that
$$\sigma(1) = \frac{\pi^2}{6}$$