令z = tan(x/2),dx = 2dz/(1 + z²),万能代换,cosx = (1 - z²)/(1 + z²)
∫ dx/(2 + cosx)
= ∫ 2/(1 + z²) * 1/[2 + (1 - z²)/(1 + z²)] dz
= ∫ 2/(1 + z²) * (1 + z²)/(2 + 2z² + 1 - z²) dz
= ∫ 2/(z² + 3) dz
= 2 * 1/√3 * arctan(z/√3) + C <== ∫ dx/(x²+a²)=(1/a)arctan(x/a)+C
= (2/√3)arctan[(1/√3)tan(x/2)] + C
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