w = ∫dy/(c-siny) 請問這怎麼積??
Set y = 2*atan(x) => dy = 2dx/(1+x^2)
=> x = tan(y/2)
=> siny = sin[2*atan(x)]
= 2*sin[atan(x)]*cos[atan(x)]
= 2x/(1+x^2)
分母 = c - siny
= c - 2x/(1+x^2)
= (c - 2x + cx^2)/(1+x^2)
= [(cx)^2 - 2cx + 1) + (c - 1)]/c(1 + x^2)
= [(cx - 1)^2 + (c - 1)]/c(1 + x^2)
w = 2c∫dx/[(cx - 1)^2 + (c - 1)]
= 2∫d(cx - 1)/[(cx - 1)^2 + (c - 1)];;; u = cx - 1
= 2∫du/[u^2 + (c - 1)]
= [2/√(c - 1)]*atan[u/√(c - 1)] + C
= [2/√(c - 1)]*atan[(cx - 1)/√(c - 1)] + C
= [2/√(c - 1)]*atan{[c*tan(y/2) - 1]/√(c - 1)} + C
= ans
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