一道定积分的题目设f（x）=1/x^3∫(x,0)sint^2dt,x不等于0（x为上限,0为下限）,在x=0处连续,求a值a,x=0

Question

一道定积分的题目
设f（x）= 1/x^3 ∫(x,0)sint^2dt,x不等于0 （x为上限,0为下限）,在x=0处连续,求a值
a,x=0

Answer 1

ƒ(x) = { [∫(0→x) sin(t²) dt]/x³,若x ≠ 0
{a,若x = 0
lim(x→0) [∫(0→x) sin(t²) dt]/x³
= lim(x→0) sin(x²)/(3x²)
= (1/3)lim(x²→0) sin(x²)/x²
= 1/3 • 1
= 1/3
∵ƒ(x)在x = 0处连续
∴a = 1/3