Question 453405
Let x = {{{pi/2 - y}}}.  Substitute for x in the integral.
 {{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}}
= {{{int((sin(pi/2 - y))^m/((sin(pi/2 - y))^m + (cos(pi/2 - y))^m),d(pi/2 - y), pi/2, 0)
}}}

= {{{- int((cosy)^m/((cosy)^m + (siny)^m),dy, pi/2, 0)
}}}
= {{{ int((cosy)^m/((cosy)^m + (siny)^m),dy, 0,pi/2)
}}} = {{{ int((cosx)^m/((cosx)^m + (sinx)^m),dx, 0,pi/2)
}}} 

==> {{{int(1, dx, 0, pi/2)}}} -  {{{ int((cosx)^m/((cosx)^m + (sinx)^m),dx, 0,pi/2)
}}}  =  {{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}} 
<==> {{{int(1, dx, 0, pi/2)}}} = {{{2*int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}} 

==> {{{(1/2)*(pi/2)}}} = {{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}}
Therefore  
{{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2) = pi/4}}}