举报
zhoumeilin3000
令u = √t,t = u²,dt = 2u du 当t = 0、u = 0 当t = 1、u = 1 ∫(0→1) arccos√t dt = ∫(0→1) arccos(u) · 2u du = ∫(0→1) arccos(u) d(u²) = u²arccos(u) - ∫(0→1) u² d(arccos(u)) <--分部积分法 = [(1)arccos(1) - 0] - ∫(0→1) u² · - 1/√(1 - u²) du = ∫(0→1) u²/√(1 - u²) du --> 令u = sinz,du = cosz dz --> 当u = 0,z = 0 当u = 1,z = π/2 = ∫(0→π/2) sin²z/cosz · cosz dz = ∫(0→π/2) sin²z dz = (1/2)∫(0→π/2) (1 - cos2z) dz = (1/2)[z - (1/2)sin2z] = (1/2)[π/2 - (1/2)sin(π)] = (1/2)(π/2 - 0) = π/4