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How to solve integration of del(sin x)dx The limits are from minus infinity to plus infinity. Please explain.

Kabyik Nandy
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  • Are you wanting to solve $\int \delta(\sin(x)),\mathrm{d}x$ or $\int \delta(x)\sin(x),\mathrm{d}x$? – Timothy Hedgeworth Aug 28 '19 at 16:08
  • Use mathjax. What is your definition of $\delta(\sin(x))$. It is obvious $\int_{-\infty}^\infty \delta(\sin(x))dx$ isn't well-defined. – reuns Aug 28 '19 at 16:22

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Using $$ δ(g(x))=\sum_{a:g(a)=0}\frac{\delta(x-a)}{|g'(a)|} $$ you would get $$ δ(\sin(x))=\sum_{k\in\Bbb Z} δ(x-k\pi). $$

Lutz Lehmann
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  • Starting from $\int δ(cx)f(x)dx=\int δ(u)f(u/c) du/|c|$, apply the linear Taylor expansion around the roots of $g$, you get a sum over all the roots of $g$ in the support of $f$. – Lutz Lehmann Aug 28 '19 at 18:18