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How to interpret this integral?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112597] How to interpret this integral?
  • From: Julian <julian.stoev at gmail.com>
  • Date: Wed, 22 Sep 2010 01:57:18 -0400 (EDT)

Hello All,

After long interruption with symbolic computing, I am struggling how
to make a useful result out of this integral.

Integrate[(aV*t + v0)*Cos[(aW*t^2)/2 + th0 + t*w0, {t, 0, Ts}]

It comes from the differential equations describing the motion of a
wheeled robot. aV and aW are accelerations and v0, w0, th0 are initial
conditions. The numerical solution clearly exists for different real
accelerations, including positive, negative and zero.

However the symbolic solution of Mathematica is:
(Sqrt[Pi]*(-(aW*v0) + aV*w0)*Cos[th0 - w0^2/(2*aW)]*
   FresnelC[w0/(Sqrt[aW]*Sqrt[Pi])] + Sqrt[Pi]*(aW*v0 - aV*w0)*
   Cos[th0 - w0^2/(2*aW)]*FresnelC[(aW*Ts + w0)/(Sqrt[aW]*Sqrt[Pi])]
+
  aV*Sqrt[aW]*(-Sin[th0] + Sin[th0 + (aW*Ts^2)/2 + Ts*w0]) +
  Sqrt[Pi]*(aW*v0 - aV*w0)*(FresnelS[w0/(Sqrt[aW]*Sqrt[Pi])] -
    FresnelS[(aW*Ts + w0)/(Sqrt[aW]*Sqrt[Pi])])*Sin[th0 - w0^2/
(2*aW)])/
 aW^(3/2)

Note that aW can be found inside a Sqrt function and also in the
denominator. While I can see that FresnelS[Infinity] is well defined,
so aW==0 should not be a real problem, it is  still very problematic
how to use this result at this particular point. The case of negative
aW is also interesting, because it will require a complex FresnelS and
FresnelC. I somehow managed to remove this problem using
ComplexExpand, but I am not sure this is the good solution.

I tried to give Assumptions -> Element[{aW, aV, w0, v0, Ts, th0},
Reals] to the integral, but there is no change in the solution.

While I understand that the solution of Mathematica is correct in the
strict mathematical sense, I have to use the result to generate a C-
code, which will be evaluated numerically and the solution I have now
is not working for this.

Can some experienced user give a good advice how use get a solution of
the problem?

Thank you in advance!

--JS


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