Question

划红线的积分式怎么转化的

求详细的转化过程，谢谢

Answer 1

let
u=x-t
du =-dt
t=0, u=x
t=x, u=0
∫(0->x) tf(t-x)dt
= ∫(x->0) (x-u)f(u) (-du)
= ∫(0->x) (x-u)f(u) du
= ∫(0->x) (x-t)f(t) dt
lim(x->0) [∫(0->x) (x-t)f(t) dt - (1/2)x^2 ]/bx^k
=lim(x->0) [x∫(0->x) f(t) dt -∫(0->x) tf(t) dt - (1/2)x^2 ]/( bx^k) (0/0)
分子，分母分别求导
=lim(x->0) [xf(x)+∫(0->x) f(t) dt -xf(x) -x ]/[ bk.x^(k-1) ]
=lim(x->0) [∫(0->x) f(t) dt -x ]/[ bk.x^(k-1) ]