       # Need integration help!

```I'm evaluating Mathematica on a friend's machine, trying to see if I can
do this calc so I can dump another math engine I own. I have only
Wolfram's book and my wits.

I am calculating an integral that looks like this:

4
/
| T(w)/(w^2 - 5^2) dw
/
2

It happens that T(w) is a previously defined, debugged, and well-behaved
InterpolatingFunction over the range of integration. I am using
NIntegrateInterpolatingFunction. The exact Mathematica coding I use is:

NIntegrateInterpolatingFunction[T[w]/(w^2 - 5^2), {w, 1, 4}]

This calculates a number. FYI, the result happens to be 0.0111 for my
T[w]. All's well so far. But, as a lead-in to my punchline question, I
am also defining a function F thusly:

F[e_]=NIntegrateInterpolatingFunction[T[w]/(w^2 - e^2), {w, 1, 4}]

When I define this, I get a message 'Integrand is not numerical at w =
2.' I'm not sure what this means. I thought it was just a warning
reminder that the integrand could be singular if I wasn't careful, but
I didn't worry about it since I am making sure my value of 'e' is
always well outside of  the integrating limits (in this case, e=5). But
now I realize this warning probably is serious and that I need to be
more careful because

F yields 0.0027, NOT the previous 0.0111 !!!!!

Can anyone explain the nuances of this subtlety? What is the deep
meaning of the above warning and how can it mess me up? Am I doing
something improperly or dumb?

Now, if that's not enough, *here* is the -punchline- question:

What I *really* need to do for my application is define

F[e_]=NIntegrateInterpolatingFunction[T[w]/(w^2 - e^2), {w, 1, e-1}]

so I can generate a list of values F, F, F[whatever]. But
Mathematica does not like the upper limit to be anything other than a
number, apparently. Regardless of my earlier question, how can I define
a function whose arguments may include an integral's upper limit?

Help! I'd like to switch to Mathematica. There's gotta be a way to do
this.

TIA

Stefan Jeglinski

```

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