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令 x = 2sinu, dx = 2cosudu
x : 1 → -1, u : π/6 → -π/6
原积分 = ∫(2sinu + 2cosu)²2cosudu (π/6→-π/6)
=8∫(sinu + cosu)²cosudu (π/6→-π/6)
=8∫(1 + 2sinucosu)cosudu (π/6→-π/6)
=8∫(1 + 2sinucosu)dsinu (π/6→-π/6)
=8∫dsinu + 16∫sinucosudsinu (π/6→-π/6)
=8∫dsinu + 16∫sinucos²udu (π/6→-π/6)
=8∫dsinu - 16∫cos²udcosu (π/6→-π/6)
=8sinu - (16/3)cos³u (π/6→-π/6)
=8(-1/2 - 1/2) - (16/3)(1/8 - 1/8)
=-8
x : 1 → -1, u : π/6 → -π/6
原积分 = ∫(2sinu + 2cosu)²2cosudu (π/6→-π/6)
=8∫(sinu + cosu)²cosudu (π/6→-π/6)
=8∫(1 + 2sinucosu)cosudu (π/6→-π/6)
=8∫(1 + 2sinucosu)dsinu (π/6→-π/6)
=8∫dsinu + 16∫sinucosudsinu (π/6→-π/6)
=8∫dsinu + 16∫sinucos²udu (π/6→-π/6)
=8∫dsinu - 16∫cos²udcosu (π/6→-π/6)
=8sinu - (16/3)cos³u (π/6→-π/6)
=8(-1/2 - 1/2) - (16/3)(1/8 - 1/8)
=-8
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