How to Compute an Integral Involving a Delta Function and Sine?

[helpcometk]

Homework Statement

Compute ∫^{∞}_{-∞}dx (x²+a²)^-1δ(sin(2x)), without calculating the resulting sum.

Homework Equations

This is a very specific integral which ,has a delta function δ operating on sin function

The Attempt at a Solution

Does anyone know this integral ? I haven't seen before any similar examples.

 

[vanhees71]

You haven't listed all relevant equations yet:
\delta[f(x)]=\sum_{k} \frac{1}{\left |f'(x_k) \right|} \delta(x-x_k),
where x_k runs over all zeros of f, which all must be simple zeros of course in order that f'(x_k) \neq 0 for all k.