`int(e^(-x/2)sqrt(1-sinx))/(1+cosx)dx`

Explanation

Putting `(-x)/(2)=t, " we get " x=-2t anddx=-2dt`.
`:.I=int(sqrt(1-sinx))/((1+cosx))e^(-x//2)dx`
`=int(sqrt(1-sin(-2t)))/({1+cos(-2t)})e^(t)(-2dt)=-2int(sqrt(1+sin2t))/((1+cos2t))e^(t)dt`
`=-2int(sqrt(cos^(2)t+sin^(2)t+2sintcost))/(2cos^(2)t)e^(t)dt`
`=-2int((cost+sint))/(2cos^(2)t)e^(t)dt=-int(sect+sect tan t)e^(t)dt`
`=-inte^(t)f(t)+f'(t)dt,"where"f(t)=sect`
`-e^(t)f(t)+C=-e^(-x//2)sec((-x)/(2))+C=-e^(-x//2)sec(x)/(2)+C`.