Not sure how to finish this integral $\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$ Dirac delta is a symmetric function defined as $$\int_{-\infty}^\infty f(t)\delta(t-A)dt = f(A)$$
Find the value of $$\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$$
SOLUTION:
Let $t=x^2$ then $dt=2xdx \rightarrow dx = dt/(2x)$ and the integral is $$\int_{-\infty}^\infty tx \delta(t-2)dt/(2x) = \int_{-\infty}^\infty t/2 \delta(t-2)dt = f(A).$$ I am not quite sure what to do now in terms of finishing the problem...

OP's integral $$\int_{\mathbb{R}}\! \mathrm{d}x~ x^3 \delta(x^2-2)~=~0$$ is zero because it's an odd integrand on a symmetric interval.
