definite integration of sin^2 x/sin x+cos x from 0 to pie

integral sin^2x/sinx+cosx from 0 to pie =

 integral{( sin^2x)/(sinx+cosx)  + (cos^2x)/(sinx+cosx)}from 0to pie/2     after applying propertie of integral

=integral( 1/sinx+cosx)integral from 0 t0 pie/2

=  put sinx=2tan(x/2)/(1+tan^x/2) and cosx=(1-tan^x/2)/(1+tan^2x/2)

now expression becomes   integral (sex^2x/2) /(1+2tan(x/2) -tan^2x/2) limit from 0 to pie/2 

now put tanx/2 = t

    we get sec^2dx=2dt 

expression becomes integral 2dt/1+2t-t^2  limit 0 to 1

 =integral -2/t^2 - 2 limit 0 to 1

 =(1/underoot2) .(log(underoot2- 1)/(underoot2 +1))