Integration of the product of a function and the Heaviside function

Question

while solving a problem in quantum electrodynamics I have to solve the following integral: $$\int_0^t\hat{\pi}(t')\int_{0}^{+\infty} k^2 \Big\{ e^{ik(r-R)} -e^{-ik(r-R)}\Big\}e^{-ick (t-t')}dk \, dt'.$$ As the integral over $k$ is similar to Fourier Transform integrals, it makes sense to write it as $$\int_{-\infty}^{+\infty} k^2 \mathcal{H}(k) e^{ick(t'-t+\frac{r-R}{c})}dk -\int_{-\infty}^{+\infty} k^2 \mathcal{H}(k) e^{ick(t'-t-\frac{r-R}{c})}dk$$ with $\mathcal{H}(k)$ the Heaviside function. I know that the Fourier Transform of the Heaviside function will have a contribution of the Dirac Delta and another of the Cauchy Principal Value. The first one makes sense to me as it will easily solve the integral over $t'$ evaluating the operator $\hat{\pi}$ in a retarded time. Furthermore, due to the $k^2$ factor I can't find a way to solve this integral. Notice that it one defines $\xi=c(t'-t-\frac{r-R}{c})$, my problem is reduced to solve the integral $$ \int_{-\infty}^{+\infty} k^2 \mathcal{H}(k) e^{i\xi k}dk $$ I hope you can help me to solve it