∫(0→π/2) dx/(1 + cos²x)
= ∫(0→π/2) dx/[1 + (1 + cos2x)/2]
= 2∫(0→π/2) dx/(3 + cos2x)，θ = 2x
= ∫(0→π) dθ/(3 + cosθ)
= ∫(0→π) dθ/[3sin²(θ/2) + 3cos²(θ/2) + cos²(θ/2) - sin²(θ/2)]
= ∫(0→π) d(θ/2)/[2cos²(θ/2) + sin²(θ/2)]
= ∫(0→π) d(θ/2)/{cos²(θ/2)[2 + tan²(θ/2)]}
= ∫(0→π) d[tan(θ/2)]/[2 + tan²(θ/2)]
= (1/√2){arctan[(1/√2)tan(θ/2)]} |(0→π)
= 1/√2 * π/2
= π/(2√2)
飞差不多
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