∫sin^8(x/2)dx

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∫sin^8(x/2)dx
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∫sin^8(x/2)dx
∫sin^8(x/2)dx

∫sin^8(x/2)dx
∫[sin(x/2)]^8dx=∫[35/128-cosx/4+7cos(2x)/32+cos(4x)/128-cosx(1-(sinx)^2)/4]dx
(应用倍角公式)
=∫[35/128-cosx/4+7cos(2x)/32+cos(4x)/128]dx-∫[cosx(1-(sinx)^2)/4]dx
=35x/128-sinx/4+7sin(2x)/64+sin(4x)/512-∫[(1-(sinx)^2)/4]d(sinx)
=35x/128-sinx/4+7sin(2x)/64+sin(4x)/512-sinx/4+(sinx)^3/12+C (C是常数).

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