ƒ(x) = (1/2)∫(0→x) (x - t)²g(t) dt
= (1/2)∫(0→x) (x² - 2 • x • t + t²)g(t) dt
= (1/2)[∫(0→x) x²g(t) dt - ∫(0→x) 2xtg(t) dt + ∫(0→x) t²g(t) dt]
= (1/2)x²∫(0→x) g(t) dt - x∫(0→x) tg(t) dt + (1/2)∫(0→x) t²g(t) dt
d/dx x²∫(0→x) g(t) dt = x² • g(x) + 2x • ∫(0→x) g(t) dt
d/dx x∫(0→x) tg(t) dt = x • xg(x) + ∫(0→x) tg(t) dt
d/dx ∫(0→x) t²g(t) dt = x²g(x)
于是ƒ'(x) = (x²/2)g(x) + x∫(0→x) g(t) dt - x²g(x) - ∫(0→x) tg(t) dt + (x²/2)g(x)
= x∫(0→x) g(t) dt - ∫(0→x) tg(t) dt
于是ƒ''(x) = x • g(x) + ∫(0→x) g(t) dt - xg(x) = ∫(0→x) g(t) dt