函数f在（0,+∞）连续,对任意给定的a∈R+,b∈R+,从a到abf（x）的积分与a无关,求证f（x）=C/xC为常数

Question

函数f在（0,+∞）连续,对任意给定的a∈R+,b∈R+,从a到ab f（x）的积分与a无关,求证f（x）=C/x
C为常数

Answer 1

g'(x) = f(x)
从a到ab f（x）的积分 = g(ab) - g(a)
g(ab) - g(a) 与a无关,
d(g(bx) - g(x))/dx = 0
g'(bx)*b - g'(x) = 0
g'(t)*t -g'(1) = 0
g'(t) = g'(1)/t
f(x) = g'(x) = C/x.(C=g'(1))