Question

求(f(x)-f(t))dt的定积分的导数

见配图，有点想不通f(x)dt的定积分的导数该怎么求

Answer 1

A(x) = ∫<a, x>f(x)dt - ∫<a, x>f(t)dt
= f(x)∫<a, x>dt - ∫<a, x>f(t)dt = (x-a)f(x) - ∫<a, x>f(t)dt
则 A'(x) = f(x) + (x-a)f'(x) - f(x) = (x-a)f'(x)