设f(x)=∫((pi,x) sintdt/t,求∫(0,pi) f(x)dx

Leibniz公式：d/dx ∫(a(x),b(x)) f(t) dt = b'(x) * f[b(x)] - a'(x) * f[a(x)]
f(x) = ∫(π,x) sint/t dt
f'(x) = x' * (sinx)/x - π' * (sinπ)/π = (sinx)/x
f(π) = ∫(π,π) sint/t dt = 0,(上限和下限相同,面积为0)
∫(0,π) f(x) dx
= xf(x) |(0,π) - ∫(0,π) x d[f(x)],分部积分法
= πf(π) - 0f(0) - ∫(0,π) x * f'(x) dx
= - ∫(0,π) x * (sinx)/x dx
= - ∫(0,π) sinx dx
= cosx |(0,π)
= cosπ - cos0
= - 1 - 1
= - 2