Antiderivative of Heaviside function with absolute-value-argument

Question

I'm looking to calculate the antiderivative of $$\Theta (R-|x|),$$where $\Theta$ denotes the Heaviside step function and $R$ is a given constant. On Wikipedia it is given that $\int_{-\infty}^x \Theta(t) \mathrm{d} t=x\Theta(x)$, however, here the argument is a bit more involved. Any comments or answers on how to go about this problem are appreciated.

Answer 1

If $R\le0$, then the antiderivative is just some constant $C$.

If $R>0$, then, by inspection, the antiderivative is $$(R+x)\theta(R-|x|)+2R\theta(x-R)+C.$$

If we take the derivative of the latter w.r.t. $x$, we find $$\theta(R-|x|)-2R\delta(R-x)\theta(x)+2R\delta(x-R)+(R+x)\delta(R+x)\theta(-x) + R\delta(x)-R\delta(-x) \\ =\theta(R-|x|).$$