f(x) = ∫(0→a - x) e^[y(2a - y)] dy
f'(x) = - e^[(a - x)(2a - (a - x))]
= - e^[(a - x)(a + x)]
= - e^(a² - x²)
∫(0→a) f(x) dx
= xf(x) |(0→a) - ∫(0→a) x d[f(x)]
= af(a) - ∫(0→a) xf'(x) dx
= - ∫(0→a) x · [- e^(a² - x²)] dx、注意f(a) = 0
= (- 1/2)∫(0→a) e^(a² - x²) d(a² - x²)
= (- 1/2)e^(a² - x²) |(0→a)
= (- 1/2)[1 - e^(a²)]
= (1/2)[e^(a²) - 1]