求不定积分如下
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As shown in the Figure:
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x*[cos(x/2)]^4/(sinx)^3
=x*[cos(x/2)]^4/[2sin(x/2)cos(x/2)]^3
= 1/8*x*cos(x/2)/[sin(x/2)]^3
= 1/8*x*cot(x/2)*csc^2(x/2)
∫ x*[cos(x/2)]^4/(sinx)^3 dx
=∫ 1/8*x*cot(x/2)*csc^2(x/2) dx
= -1/16*∫ x*cot(x/2) dcot(x/2)
= -1/32*∫ x dcot²(x/2)
= -1/32*{ xcot²(x/2) - ∫ cot²(x/2) dx }
= -1/32*{ xcot²(x/2) - ∫ [csc²(x/2) - 1] dx }
= -1/32*{ xcot²(x/2) + 1/2*cot(x/2) + x ] + C
=x*[cos(x/2)]^4/[2sin(x/2)cos(x/2)]^3
= 1/8*x*cos(x/2)/[sin(x/2)]^3
= 1/8*x*cot(x/2)*csc^2(x/2)
∫ x*[cos(x/2)]^4/(sinx)^3 dx
=∫ 1/8*x*cot(x/2)*csc^2(x/2) dx
= -1/16*∫ x*cot(x/2) dcot(x/2)
= -1/32*∫ x dcot²(x/2)
= -1/32*{ xcot²(x/2) - ∫ cot²(x/2) dx }
= -1/32*{ xcot²(x/2) - ∫ [csc²(x/2) - 1] dx }
= -1/32*{ xcot²(x/2) + 1/2*cot(x/2) + x ] + C
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