设f(x)一阶可微,y=∫(0积到x^2)xf(t)dt,试求d^2y/dx^2

Question

设f(x)一阶可微,y=∫(0积到x^2) xf(t)dt,试求d^2y/dx^2

Answer 1

y=∫(0->x^2) xf(t)dt
y' = xd/dx(∫(0->x^2) f(t)dt) + ∫(0->x^2) f(t)dt
= 2x^2.f(x^2) +∫(0->x^2) f(t)dt
y'' = 2[ 2xf(x^2) + 2x^3. f'(x^2)] + 2xf(x^2)
=6xf(x^2) +4x^3.f'(x^2)