?(x+sinx)/(1+cosx)dx =

∫ ( 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

BOOK

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