Proceed as follows .,
Integral sqrt sin2 x/2 + cos2 x/2 -2sin x/2 *cos x/2 * e-x/2 dx / 2cos2x/2
Integral sqrt (sinx/2 – cosx/2)2 * e-x/2 dx / 2cos2x/2
Integral sin x/2 – cos x/2 * e-x/2 dx
Integral e-x/2 [1/2 tan x/2 *secx/2 – 1/2 secx/2 ] dx
½ * Integral e-x/2 tanx/2 *sec x/2 dx - ½ integral e-x/2 secx/2 dx
Using integration by parts ,
1/2 integral e-x/2 tanx/2 secx/2 – ½ [ -2 e-x/2 secx/2 – integral -2 e-x/2 * ½ secx/2 tanx/2 dx ]