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求ln^3(x)/x^3的原函数.
题目详情
求ln^3(x)/x^3的原函数.
▼优质解答
答案和解析
∫ (lnx)^3/x^3 dx
=-(1/2)∫ (lnx)^3d(1/x^2)
=-(1/2)(lnx)^3/x^2 +(3/2)∫ (lnx)^2/x^3 dx
=-(1/2)(lnx)^3/x^2 -(3/4)∫ (lnx)^2 d(1/x^2)
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 +(3/2)∫ (lnx)/x^3 dx
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)∫ (lnx) d(1/x^2)
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)(lnx)/x^2 + (3/4)∫ (1/x^3)dx
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)(lnx)/x^2 - (3/8) (1/x^2) + C
=-(1/2)∫ (lnx)^3d(1/x^2)
=-(1/2)(lnx)^3/x^2 +(3/2)∫ (lnx)^2/x^3 dx
=-(1/2)(lnx)^3/x^2 -(3/4)∫ (lnx)^2 d(1/x^2)
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 +(3/2)∫ (lnx)/x^3 dx
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)∫ (lnx) d(1/x^2)
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)(lnx)/x^2 + (3/4)∫ (1/x^3)dx
=-(1/2)(lnx)^3/x^2 -(3/4) (lnx)^2/x^2 -(3/4)(lnx)/x^2 - (3/8) (1/x^2) + C
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