Not sure how to finish this integral $\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$
Solution 1:
You can use the property
$$ \int_{-\infty}^{+\infty}f(x)\delta(g(x))~{\rm d}x = \sum_i \frac{f(x_i)}{|g'(x_i)|} $$
where $x_i$ are the roots of $g$: $g(x_i) = 0$. In your case you have $g(x) = (x - 2^{1/2})(x + 2^{1/2})$, $x_1 = 2^{1/2}$ and $x_2 = -2^{1/2}$, $g'(x) = 2x$ and $f(x) = x^3$.
Putting everything together
\begin{eqnarray} \int_{-\infty}^{+\infty}x^3\delta(x^2 - 2)~{\rm d}x = \frac{(2^{1/2})^{3}}{|2(2^{1/2})|} + \frac{(-2^{1/2})^{3}}{|2(-2^{1/2})|} = \cdots \end{eqnarray}
can you take it from here?