设f(x)在(-无穷,+无穷)内连续,证明(d/dx)∫(0~x)(x-t)f'(t)dt=f(x)-f(a)

(d/dx)∫(0~x)(x-t)f'(t)dt=(d/dx)[x∫(0~x)f'(t)dt-∫(0~x)tf'(t)dt]
= ∫(0~x)f'(t)dt + x*f'(x) -x*f'(x)=∫(0~x)f'(t)dt =f(x)-f(0)