求x-(a+b)绝对值的不定积分,紧急,

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求x-(a+b)绝对值的不定积分,紧急,

求x-(a+b)绝对值的不定积分,紧急,
求x-(a+b)绝对值的不定积分,紧急,

求x-(a+b)绝对值的不定积分,紧急,
1/2x²-(a+b)x+C (x>=a+b)
(a+b)x-1/2x²+C (x

  若x>=a+b,
∫|x-(a+b)|dx = ∫[x-(a+b)]dx = (1/2)x^2 - (a+b)x + C1,
若x ∫|x-(a+b)|dx = ∫[(a+b)-x]dx = (a+b)x - (1/2)x^2 + C2,
记该不定积分(原函数)为F(x),由于原函数是连续的,应有

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  若x>=a+b,
∫|x-(a+b)|dx = ∫[x-(a+b)]dx = (1/2)x^2 - (a+b)x + C1,
若x ∫|x-(a+b)|dx = ∫[(a+b)-x]dx = (a+b)x - (1/2)x^2 + C2,
记该不定积分(原函数)为F(x),由于原函数是连续的,应有
F((a+b)-0) = F((a+b)+0) = F(a+b) ,
可得 C1 = (a+b)^2+C2,
于是,记 C2 = C,则
F(x) = (1/2)x^2 - (a+b)x + (a+b)^2 + C,x>=a+b,
= (a+b)x - (1/2)x^2 + C, x

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