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Topic: Re: An analytical solution to an integral not currently in
Replies: 0

 Daniel Posts: 59 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 non-zero 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 (2013-07-13): "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*x-b)*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)
>