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MathGroup Archive 2006

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Re: Why does this lead to an answer with complex numbers?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71472] Re: Why does this lead to an answer with complex numbers?
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 20 Nov 2006 02:43:54 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <ejosmm$n3k$1@smc.vnet.net>

aaronfude at gmail.com wrote:
> The expression is
> 
> \!\(FullSimplify[
>     Assuming[\[Beta] > 0 && \[Beta] < Pi/2,
>       Integrate[\(-Log[\@\(1 + x\^2\) - 1/11*x\ ]\), \ x]]]\)

Two remarks: the integral is complex and it is independent of any 
variable or constant called beta.

FullSimplify[Assuming[\[Beta] > 0 && \[Beta] < Pi/2,
  Integrate[-Log[Sqrt[1 + x^2] - (1/11)*x], x]]]

returns
             x               2
x - x Log[-(--) + Sqrt[1 + x ]] -
             11

       1                    2 Sqrt[30] x
   ---------- (11 (4 ArcTan[------------] +
   8 Sqrt[30]                    11

                                      2
        4 ArcTan[2 Sqrt[30] Sqrt[1 + x ]] +

                                 2 2
        I (2 Log[900 (121 + 120 x ) ] -

                           2
           Log[(121 + 120 x )

                          2                  2
             (-121 - 122 x  + 22 x Sqrt[1 + x ])] -

                           2
           Log[(121 + 120 x )

                         2                  2
             (121 + 122 x  + 22 x Sqrt[1 + x ])])))


Integrate[-Log[Sqrt[1 + x^2] - (1/11)*x], x]

yields

  1                              2 Sqrt[30] x
--- (240 x - 44 Sqrt[30] ArcTan[------------] -
240                                  11

                                             2
     44 Sqrt[30] ArcTan[2 Sqrt[30] Sqrt[1 + x ]] -

                                       2 2
     22 I Sqrt[30] Log[900 (121 + 120 x ) ] -

                 x               2
     240 x Log[-(--) + Sqrt[1 + x ]] +
                 11

                                   2
     11 I Sqrt[30] Log[(121 + 120 x )

                     2                  2
        (-121 - 122 x  + 22 x Sqrt[1 + x ])] +

                                   2
     11 I Sqrt[30] Log[(121 + 120 x )

                    2                  2
        (121 + 122 x  + 22 x Sqrt[1 + x ])])

FreeQ[%, \[Beta]]

--> True

Regards,
Jean-Marc


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