微分積分～立命館・理系全学部・2019.2.2数学Ⅲ(2)

[3]   K - L = ∫ log(tanx) dx [π/2-b, b] = ∫ log(1/tany) -dy [b, π/2-b] = -∫ log(tany) dy [π/2-b, b] = -(K-L)  ∴K-L=0

       K + L = ∫ log(sinx cosx) dx = ∫ log(1/2 sin2x) dx = ∫ log(sint) - log2 1/2 dt [π-2b, 2b]

                 = ∫ log(sint) - log2 dt [π/2, 2b] = ∫ log(sint) dt [π/2, 2b] + (2b - π/2) log2

       ∴ K → 1/2 J - π/4 log2　(b→+0)

[4]  J = K  ∴ J = -π/2 log2

      I(a) = π log2 - a log(1-cosa) - (π-a)log2 - 4∫ log(sinx) dx [π/2 a/2]

　　　→ -π/2 log2 - 0 - π log2 - 4(-π/2 log2) = 2π log2