Question

求函数:t乘|x-t| 在[0，1]的定积分为y，求函数y。注:定积分表达式后面是dt，不是dx

Answer 1

y(x)=∫[0→1] t|x-t| dt
1、当x<0时，此时x<t
y(x)=∫[0→1] t|x-t| dt
=∫[0→1] t(t-x) dt
=∫[0→1] (t²-tx) dt
=(1/3)t³-(1/2)t²x |[0→1]
=(1/3)-(1/2)x

2、当0≤x≤1时
y(x)=∫[0→1] t|x-t| dt
=∫[0→x] t(x-t) dt + ∫[x→1] t(t-x) dt
=∫[0→x] (tx-t²) dt + ∫[x→1] (t²-tx) dt
=(1/2)t²x-(1/3)t³ |[0→x] + (1/3)t³-(1/2)t²x |[x→1]
=(1/6)x³+(1/3)-(1/2)x+(1/6)x³
=(1/3)-(1/2)x+(1/3)x³

3、当x>1时，此时x>t
y(x)=∫[0→1] t|x-t| dt
=∫[0→1] t(x-t) dt
=(1/2)x-(1/3)

综上：f(x)=(1/3)-(1/2)x x<0
(1/3)-(1/2)x+(1/3)x³ 0≤x≤1
(1/2)x-(1/3) x>1

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Answer 2

分段讨论积分