※ 引述《euler3002 (無)》之銘言： : S dx/1+tanx^(根號二)， 範圍是0到pi/2 : 完全不知道如何下手，謝謝 π/2 1 ∫ ------------------ dx 0 1 + (tanx)^(√2) π/2 1 = ∫ --------------------------- dx 0 1 + ((sinx)/(cosx))^(√2) π/2 (cosx)^(√2) = ∫ ----------------------------- dx 0 (cosx)^(√2) + (sinx)^(√2) π/2 1 π/2 (cosx)^(√2) 令 I = ∫ ------------------ dx = ∫ ----------------------------- dx 0 1 + (tanx)^(√2) 0 (cosx)^(√2) + (sinx)^(√2) π/2 (cosx)^(√2) 則 I = ∫ ----------------------------- dx 0 (cosx)^(√2) + (sinx)^(√2) 0 (cos(π/2 - y))^(√2) = ∫ (-----------------------------------------------)(-1) dy π/2 (cos(π/2 - y))^(√2) + (sin(π/2 - y))^(√2) (令 y = π/2 - x , 則 x = π/2 - y , dx = - dy) (x = 0 => y = π/2 , x = π/2 => y = 0) π/2 (siny)^(√2) = ∫ ----------------------------- dy 0 (siny)^(√2) + (cosy)^(√2) π/2 (sinx)^(√2) = ∫ ----------------------------- dx 0 (sinx)^(√2) + (cosx)^(√2) π/2 (sinx)^(√2) 則 2I = ∫ ----------------------------- dx 0 (sinx)^(√2) + (cosx)^(√2) π/2 (cosx)^(√2) + ∫ ----------------------------- dx 0 (sinx)^(√2) + (cosx)^(√2) π/2 (sinx)^(√2) + (cosx)^(√2) = ∫ ----------------------------- dx 0 (sinx)^(√2) + (cosx)^(√2) π/2 |π/2 π = ∫ 1 dx = x | = ---- 0 |0 2 π π/2 1 π I = --- => ∫ ------------------ dx = ---- 4 0 1 + (tanx)^(√2) 4 --

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