∫ ( x + sin(x)) dx /(1 + cos(x)
= ∫ ( x + 2sin(x/2)cos(x/2)) dx /(1 + 2cos^2(x/2) – 1)
= ∫ ( x + 2sin(x/2)cos(x/2)) dx / 2cos^2(x/2)
= ∫ x dx / 2 cos^2(x/2) + ∫ tan(x/2) dx
= 1/2∫ x sec^2(x/2) dx + ∫ tan(x/2) dx ——————–(1)
integrate the first integral by parts
u = x
du = dx
dv = (1/2)sec^2(x/2) dx
v = tan(x/2)
1/2∫ x sec^2(x/2) dx = x tan(x/2) – ∫ tan(x/2) dx
substitute in (1)
1/2∫ x sec^2(x/2) dx + ∫ tan(x/2) dx = x tan(x/2) – ∫ tan(x/2) dx + ∫ tan(x/2) dx
= x tan(x/2) + C
