^k}\mathrm{d}x)
looks like one of those almost-friendly integrals that end up being analytically untractable. I'd play around with the exponential integral function, or some weighted average thereof, since
 + C.)
Anyway, I like your approach. Have you checked your your first integral numerically (for reasonable values of
)
and
in at least two dimensions), or just the last one? The first one looks fine, but I tend to become particularly wary when integrating wrt another variable can create a stochastic integral.
My other, dumb 10:15-AM-with-no-sleep guess for a possible culprit is that
is a too hasty conclusion from the obvious probabilities for each direction above.
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
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