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Daniel
Posts:
58
Registered:
7/24/12


Re: An analytical solution to an integral not currently in
Posted:
Jul 16, 2013 5:54 AM


First I have to say that "Another system's unconfirmed answer" is not a good enough reason for such a topic title.
For the math: The "another system answer" is correct only for a=0. And Mathematica's Integrate[] gives the same answer up to a constant.
However, for nonzero a, the given analytical expression is not correct, as can be seen by plotting the following:
f[a_, b_, x1_, x2_] := NIntegrate[1/Sqrt[Log[x] + a x + b], {x, x1, x2}]
g[a_, b_, x_] := Sqrt[\[Pi]] I Exp[a x  b] Erf[I Sqrt[Log[x] + a x + b]]
Plot[{f[1, 0, 1, x], g[1, 0, x]  g[1, 0, 1]}, {x, 1, 25}]
Plotting for a=0 will show identity:
Plot[{f[0, 1, 1, x], g[0, 1, x]  g[0, 1, 1]}, {x, 1, 25}]
> Question: Integral dx of 1/sqrt(Log[x] + a*x + b) > (sorry if my notation is off; I just used the online > integrator and don't have Mathematica proper, > http://integrals.wolfram.com/index.jsp?expr=1%2Fsqrt%2 > 8Log%5Bx%5D+%2B+a*x+%2B+b%29) > (the online integrator returned this as of the time > of writing this (20130713): "Mathematica could not > find a formula for your integral. Most likely this > means that no formula exists." ) > > > Another system's unconfirmed answer (in that > notation; sorry) (version 5.27.0): > sqrt(%pi)*%i*%e^(a*xb)*erf(%i*sqrt(log(x)+a*x+b)) > > Strangely, the other system only produces this result > when given, say, x(t) in all places for x (including > variable of integration). > > I can't seem to get the other system to verify its > result symbolically, but when I try random numerical > sampling, it does seem to agree, albeit horribly > plagued by floating point errors for large x. > > > Can anyone offer insight, or possibly prove it's > correctness or incorrectness? :) > > > (P.S. I just joined this group, so apologies if it's > the wrong one or I'm not following guidelines) >



