integrate(1/(sin(x)^2+1),x,a,b)?

• Subject: integrate(1/(sin(x)^2+1),x,a,b)?
• From: Raymond Toy
• Date: Wed, 27 Jun 2007 15:29:13 -0400

>>>>> "Barton" == Barton Willis <willisb at unk.edu> writes:

    Barton> Here is what I get using Maple 10
[snip]
    >> integrate(1/(sin(x)^2+1),x);
    >> 

    Barton>                           1/2                1/2
    Barton>                      1/2 2    arctan(tan(x) 2   )

    >> _EnvAllSolutions := true:
    >> integrate(1/(sin(x)^2+1),x=0..z);
    >> 

    Barton>   {
    Barton>   {      1/2 /               1/2         -Pi + 2 z    \          Pi
    Barton>   { 1/2 2    |arctan(tan(z) 2   ) + ceil(---------) Pi| , z < - ----
    Barton>   {          \                             2 Pi       /          2

    Barton>               1/2
    Barton>           Pi 2             Pi
    Barton>         - ------- , z = - ----
    Barton>              4             2

    Barton>                            1/2   1/2        Pi
    Barton>         1/2 arctan(tan(z) 2   ) 2    , z < ----
    Barton>                                             2

    Barton>             1/2
    Barton>         Pi 2           Pi
    Barton>         ------- , z = ----
    Barton>            4           2

    Barton>              1/2 /               1/2             -Pi + 2 z      \
    Barton>         1/2 2    |arctan(tan(z) 2   ) + Pi floor(---------) + Pi| ,
    Barton>                  \                                 2 Pi         /

    Barton>          Pi
    Barton>         ---- < z
    Barton>          2


I'm sorry, I can't understand what this is trying to tell me.  I see
there are 3 possible answers?  But what are those z things?

Ray

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