∫(0,a)dx/(x+√(a^2-x^2))dx
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∫(0,a)dx/(x+√(a^2-x^2))dx∫(0,a)dx/(x+√(a^2-x^2))dx∫(0,a)dx/(x+√(a^2-x^2))dx令x=asinθ,dx=acosθdθ,原式=∫(
∫(0,a)dx/(x+√(a^2-x^2))dx
∫(0,a)dx/(x+√(a^2-x^2))dx
∫(0,a)dx/(x+√(a^2-x^2))dx
令x = asinθ,dx = acosθdθ,原式= ∫(0→π/2) (acosθ)/(asinθ + acosθ) dθ,= (1/2)∫(0→π/2) 2cosθ/(sinθ + cosθ) dθ,= (1/2)∫(0→π/2) [(sinθ + cosθ) - (sinθ - cosθ)]/(sinθ + cosθ) dθ,= (1/2)∫(0→π/2) dθ - (1/2)∫(0→π/2) (sinθ - cosθ)/(sinθ + cosθ) dθ,= (1/2)(π/2) - (1/2)∫(0→π/2) - d(cosθ + sinθ)/(sinθ + cosθ) dθ,= π/4 + (1/2)ln(sinθ + cosθ) |(0→π/2),= π/4 + (1/2)[ln(1 + 0) - ln(0 + 1)],= π/4.
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