Re: Boole does not work with symbolic limits in Integrate ?

• To: mathgroup at smc.vnet.net
• Subject: [mg73122] Re: Boole does not work with symbolic limits in Integrate ?
• From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
• Date: Sun, 4 Feb 2007 06:33:10 -0500 (EST)
• References: <eq1ihd\$2ug\$1@smc.vnet.net>

```Mitch Murphy <mitch at mix5.com> wrote:
> hey hey
>
> use of the boole indicator function inside integrate only works for
> numerical arguments, symbolic limits of integration just returns the
> unevaluated integral.
>
> simple example ...
>
>         Integrate[2 u Boole[a < u], {u, 0, x}]
>
>         out[] = Integrate[2 u Boole[a < u], {u, 0, x}]
>
> i would expect ...
>
>         Integrate[2 u , {u, a, x}]
>
>         out[] = x^2 - a^2

If that's what you'd expect, then you made some implicit assumptions.
For example, Integrate[2 u Boole[a < u], {u, 0, x}] is not equal to
x^2 - a^2 if a < 0 and x > 0.

To get what you'd expect from Mathematica, just make your assumptions
explicit:

In[5]:=
Integrate[2 u Boole[a < u], {u, 0, x}, Assumptions->(a > 0 && x > a)]

Out[5]=
-a^2 + x^2

David

> but it does work with numerical limits of integration ...
>
>         Integrate[2 u Boole[a < u], {u, 0, 2}]
>
>         out[] = Piecewise[{{4, a <= 0}, {4 - a^2, 0 < a < 2 }]
>
> anybody know a trick to make this work ? i've tried a bunch of
> variations using evaluate[], logicalexpand[], piecewiseexpand[],
> assumptions->x Reals, ..., but nothing works.
>
> cheers,
> Mitch

```

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