Question

若函数f(x)具有连续的导数,则d/dx∫上限是x下限是0 (x-t)f '(t)dt=?

Answer 1

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+xd/dx∫[0,x] f '(t)dt-d/dx∫[0,x] tf '(t)dt

=f(x)+xf'(x)-xf'(x)

=f(x)