Stuck with integral $\int_{-\infty}^\infty \left( \frac{\sin(a t+b)}{at+b} \right)^2 \, dt$ I am stuck with the following integral:
$$\int_{-\infty}^\infty \left( \frac{\sin(a t+b)}{at+b} \right)^2 \, dt$$
I would like to show that $\varphi(t)=\frac{\sin(at+b)}{at+b}$ belongs to $L^2(\mathbb{R})$ and/or $L^1(\mathbb{R})$, i.e. $\int_{-\infty}^\infty | \varphi |^2 \,dt < \infty$   and/or $\int_{-\infty}^\infty | \varphi | \,dt < \infty $.
So far, I know that it is $|\frac{\sin(at+b)}{at+b}| \leq |\frac{1}{at+b}|$, but as $\int_{-\infty}^\infty |\frac{1}{at+b}|^2 \, dt$ does not converge, I cannot be conclusive. Looking at the plot it can be stated that it converges and then $\varphi \in L^2(\mathbb{R})$.

Yet another strategy: once the original integral has been reduced to $\frac{2}{|a|}\int_{0}^{+\infty}\frac{\sin^2 x}{x^2}\,dx$, one may invoke $\mathcal{L}(\sin^2 x)(s)=\frac{2}{s(4+s^2)}$ and $\mathcal{L}^{-1}\left(\frac{1}{x^2}\right)(s)=s$ to further reduce it to
$$ \frac{4}{|a|}\int_{0}^{+\infty}\frac{ds}{s^2+4} = \frac{\pi}{|a|}.$$
