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上限是4/π,下限是0,求∫(cosθ)^-3dθ
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上限是4/π,下限是0,求∫(cosθ)^-3dθ
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答案和解析
∫(0~π/4) (cosθ)⁻³ dθ
= ∫(0~π/4) sec³θ dθ
= ∫(0~π/4) secθ dtanθ
= secθtanθ |_0^π/4 - ∫(0~π/4) tanθ dsecθ
= √2 - ∫(0~π/4) tanθ * secθtanθ dθ
= √2 - ∫(0~π/4) (sec²θ - 1)secθ dθ
2∫(0~π/4) sec³θ dθ = √2 + ln|secθ + tanθ|_0^π/4
=> ∫(0~π/4) sec³θ dθ = [√2 + ln(1 + √2)]/2 ≈ 1.14779
= ∫(0~π/4) sec³θ dθ
= ∫(0~π/4) secθ dtanθ
= secθtanθ |_0^π/4 - ∫(0~π/4) tanθ dsecθ
= √2 - ∫(0~π/4) tanθ * secθtanθ dθ
= √2 - ∫(0~π/4) (sec²θ - 1)secθ dθ
2∫(0~π/4) sec³θ dθ = √2 + ln|secθ + tanθ|_0^π/4
=> ∫(0~π/4) sec³θ dθ = [√2 + ln(1 + √2)]/2 ≈ 1.14779
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