So I saw a post concerning the limit of $\frac{\sin(\sin x)}{x}$ as $x\rightarrow0$. So I graphed the function and I wondered what is its integral from $0$ to $\infty$? It seems like it is $\frac{\pi}2$ (since it oscillates less than $\text{sinc}$) but I don't know for sure. I tried the Feynman technique by setting $$I(a)=\int_{-\infty}^\infty\frac{\sin(\sin(ax))}{x}dx$$But then I ended up with a tedious integral. I think if we do $$I(a)=\int_{-\infty}^\infty\frac{\sin(a\sin(x))}{x}dx$$instead then maybe we could get something. I will type my edits later since I don't have time.
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Your function is even, so just double the value found in the linked question above^^ – Captain Chicky Jan 19 '23 at 08:18
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Yet another instance. – metamorphy Jan 19 '23 at 11:28