integrate sqrt(1 - sin x) e^(-x/2) / (1 + cos x)

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 Proceed as follows .,

Integral sqrt sinx/2 + cosx/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 ]

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