作业帮高数积分上限函数
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/29 08:11:17
高数积分上限函数
高数积分上限函数
高数积分上限函数
F(x) = ∫(0->x) xf'(t)dt ; x≠0
=0 ; x=0
∫ (0->x) xf'(t)dt
= ∫(0->x) xdf(t)
=x[f(t)](0->x)
=x[ f(x) - f(0) ]
=xf(x)
lim(x->0) F(x)
=lim(x->0)xf(x)
=0
f(0) cts
F'(0) =lim(h->0)[ F(h) -F(0)] /h
=lim(h->0)hf(h)/h
= f(0)
x≠0
F'(x) = lim(h->0) [F(x+h) - F(x)]/ h
=lim(h->0)[ ∫(0->x+h) (x+h) f'(t)dt - ∫(0->x) xf'(t)dt ] /h
=lim(h->0) [ (x+h)f(x+h) -xf(x) ] /h (0/0)
=lim(h->0) [ (x+h)f'(x+h) + f(x+h) ]
=xf'(x) + f(x)
F'(x) cts