一道定积分题b设f(x)可导,f(0)=0,F(x)=S[0到x]t^(n-1)f(x^n-t^n)dt,试计算lim(x趋近0)F(x)/x^(2n)

Question

一道定积分题b
设f(x)可导,f(0)=0,F(x)=S[0到x]t^(n-1)f(x^n-t^n)dt,试计算lim(x趋近0)F(x)/x^(2n)

Answer 1

F(x)＝∫(0～x)t^(n-1)f(x^n-t^n)dt
＝－1/n×∫(0～x) f(x^n-t^n)d(x^n-t^n) 换x^n-t^n为u
＝－1/n×∫(x^n～0) f(u)du
＝1/n×∫(0～x^n) f(u)du
lim(x→0) F(x)/x^(2n)
＝lim(x→0) F'(x)/[2n×x^(2n-1)]
＝lim(x→0) [1/n×f(x^n)×nx^(n-1)]/[2n×x^(2n-1)]
＝1/(2n)×lim(x→0) f(x^n)/x^n
＝1/(2n)×lim(x→0) [f(x^n)－f(0)]/x^n＝1/(2n)×f'(0)