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∫(x + sinx)/(1 + cosx)dx
=∫(x)/(1-cos(x)) dx+∫sin(x)/(1-cos(x))dx
=I1 +I2
I1=∫(x)/(2cos2(x/2)) dx
=0.5∫(x)*sec2(x/2) dx ...........(use byparts, x as differential and sec2(x/2) as integral)
=0.5[2*x*tan(x/2) + ∫(1)*2*tan(x/2) dx]
=0.5[2*x*tan(x/2) + 2*(- 2* ln(|cos(x/2)|) )]
=xtan(x/2)-2 ln(|cos(x/2)|) +c
I2=∫(2*sin(x/2)*cos(x/2))/(2*cos2(x/2)) dx
=∫tan(x/2) dx
=-2ln(|cos(x/2)|) + c
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