F(x)= ∫[0,x] (2t-x)f(t) dt = ∫[0,x] 2t f(t) dt - x * ∫[0,x] f(t) dt
F(-x) = ∫[0,-x] 2t f(t) dt + x * ∫[0,-x] f(t) dt 换元,令 u= -t,dt = -du
= ∫[0,x] 2(-u) f(-u) (-du) + x * ∫[0,x] f(-u) (-du) f(-u) = f(u)
= ∫[0,x] 2u f(u) du - x * ∫[0,x] f(u) du
= F(x)
即证.