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RE: Integrating UnitSteps

  • To: mathgroup at
  • Subject: [mg48806] RE: [mg48779] Integrating UnitSteps
  • From: "David Park" <djmp at>
  • Date: Thu, 17 Jun 2004 04:07:18 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

Use the Assumptions option in Integrate.

Integrate[UnitStep[x - 1]/x^2, {x, b, Infinity}, Assumptions -> b > 1]

David Park
djmp at 

From: BZ [mailto:BZ at]
To: mathgroup at

Hi guys!

I'm trying to integrate a function that has a discontinuity at a
single point.  I'm using UnitStep to do this, but it doesn't work very
well.  To illustrate this, a simple example (my real function is much
more complicated than this):

In[1]:= Integrate[1/x^2, {x, b, Infinity}]

Out[1]= -

Ok, so far so good, but now let's add a discontinuity at x=1:

In[2]:= Integrate[UnitStep[x - 1]/x^2, {x, b, Infinity}]

                               UnitStep[-1 + x]
Out[2]= If[b < 1, 1, Integrate[----------------, {x, b, Infinity}]]

Which is correct, in principle.  However, I'm trying to get an
explicit expression for b>1:

In[3]:= FullSimplify[%, b > 1]

                               UnitStep[-1 + x]
Out[3]= If[b < 1, 1, Integrate[----------------, {x, b, Infinity}]]
Why isn't this expression simplified?  Why doesn't Mathematica
evaluate the Integration inside the If[] (the UnitStep is 1 there
anyway)?  Should I be using UnitSteps at all for these kinds of


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