How can I compute the integral of the product of two Dirac delta and a polynomial?

Question

$$\int_{0}^{\infty}dk~k^{d-2}\delta(k-a)\delta(k-b).$$

I tried substituting $k^{d-2}\delta(k-a)$ with other espressions such as $\frac{d}{dk}\biggl[k^{d-2}\Theta(k-a)\biggr]-(d-2)k^{d-3}\Theta(k-a)$ and integrating by parts but it doesn't lead to the solution.

Answer 1

OP's distribution simplifies to $$\int_{\mathbb{R}}\mathrm{d}k~\theta(k)~k^{d-2}~\delta(a-k)~\delta(k-b)~=~\theta(a)~a^{d-2}~\delta(a-b),$$ cf. e.g. this related Math.SE post.

Answer 2

The contribution of the integrand is zero wherever $k\ne a$ or $k \ne b$. If $a\ne b$, then the integral is zero.

If $a=b \ge 0$, then the integral is equal to $b^{d-2}$ else zero. Thanks to @Qmechanic for pointing that out (essentially a step function multiplying this term).