Question

高等数学，定积分

高等数学，定积分我用原函数存在定理，哪里错了？

Answer 1

求 F'(x) 出错。应这样解：
F(x) = ∫<下0, 上x>t f(x^2-t^2)dt = (-1/2)∫<下0, 上x>f(x^2-t^2)d(x^2-t^2)
令 u = x^2 - t^2, 则
F(x) = (-1/2)∫<下x^2, 上0>f(u)du = (1/2)∫<下0, 上x^2>f(u)du, F(0) = 0,
则 F'(x) = 2xf(x^2), F'(0) = 0, 又 f(0) = 0, f'(0) = 1,
则 lim<x→0>F(x)/x^4 (0/0)
= lim<x→0>F'(x)/(4x^3) = lim<x→0>f(x^2)/(2x^2) (0/0)
= lim<x→0>2xf'(x^2)/(4x) = = lim<x→0>f'(x^2)/2 = 1/2