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- Homework Statement
- Find the antiderivative of ##\Theta (R-|x|)##, where ##\Theta## is the Heaviside step function and ##R## is a given constant.

- Relevant Equations
- The derivative of ##\Theta## is the Dirac delta function ##\delta## and ##\frac{x}{|x|}=\Theta(x) -\Theta(-x)##.

For ##R<0##, the antiderivative is just a constant, since then ##R-|x|## is negative for all values of ##x##, which in turn implies ##\Theta(R-|x|)## is zero for all values of ##x##. For ##R\geq 0##, and by inspection apparently, the antiderivative is

##(R+x)\Theta(R-|x|)+2R\Theta(x-R)+C.##

I'd like to confirm this is really the antiderivative by computing the derivative. I get

##-R\frac{x}{|x|}\delta(R-|x|)+\Theta(R-|x|)-x \frac{x}{|x|}\delta(R-|x|)+2R\delta(x-R).##

Using the identity ##\frac{x}{|x|}=\Theta(x) -\Theta(-x)##, one can simplify further

##-R\Theta(x)\delta(R-|x|)+R\Theta(-x)\delta(R-|x|)+\Theta(R-|x|)-x \Theta(x)\delta(R-|x|)+x \Theta(-x)\delta(R-|x|)+2R\delta(x-R).##

This should be possible to simplify even further, although I am stuck here. Any help is appreciated.