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

微分積分～立命館・理系全学部・2019.2.2数学Ⅲ(1)

[1]  f(t) = t sint (1+ cost) / (1 - cost)(1 + cost) = t / sint x (1+cost) → 1 x 2 = 2

[2]  イ: sin t / 1 - cos t

      f = t          g = log ( 1 - cost )

      f' = 1        g' = sint / 1 - cost

      ウ: π log 2

　  ∫ log(1-cost) dt = ∫ log(1-cos2x) 2dx = 2 ∫ log(2sin²x) dx = 2x log2 + 4 ∫ log(sinx)dx

      エ: 2(π/2-a/2)log2 = (π-a)log2