$\int_{-2\pi}^{2\pi}(1−u_0(x))\sin(x/2)(\delta(x + π) + \delta(x−π))\mathrm{d}x$

Question

Im not sure where to begin with this, or what formula or theory is being tested. any suggestions to what method I should be using?

Answer 1

Hint: you can use this property of $\delta$: $\displaystyle\int_{-a}^af(x)\delta(x-t)dx=f(t)$.

Answer 2

Break the integral in two using linearity, and find that $$ \int_{-2\pi}^{2\pi}(1-u_0(x))\sin(x)\delta(x-\pi)\mathrm dx=(1-u_0(\pi))\sin(\pi)=0 $$ the other piece is similar.