∫sin(lnx)/x^3dx
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令A=∫sin(lnx)/x^3dx
=(-1/2)*∫sin(lnx)d(1/x^2)
=(-1/2)*sin(lnx)/x^2+(1/2)*∫(1/x^2)d[sin(lnx)]
=(-1/2)*sin(lnx)/x^2+(1/2)*∫cos(lnx)/x^3dx
=(-1/2)*sin(lnx)/x^2-(1/4)*∫cos(lnx)d(1/x^2)
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2+(1/4)*∫(1/x^2)d[cos(lnx)]
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2-(1/4)*∫sin(lnx)/x^3dx
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2-(1/4)*A
4A=-2sin(lnx)/x^2-cos(lnx)/x^2-A
A=(-2/5)*sin(lnx)/x^2-(1/5)*cos(lnx)/x^2+C，其中C是任意常数
=(-1/2)*∫sin(lnx)d(1/x^2)
=(-1/2)*sin(lnx)/x^2+(1/2)*∫(1/x^2)d[sin(lnx)]
=(-1/2)*sin(lnx)/x^2+(1/2)*∫cos(lnx)/x^3dx
=(-1/2)*sin(lnx)/x^2-(1/4)*∫cos(lnx)d(1/x^2)
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2+(1/4)*∫(1/x^2)d[cos(lnx)]
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2-(1/4)*∫sin(lnx)/x^3dx
=(-1/2)*sin(lnx)/x^2-(1/4)*cos(lnx)/x^2-(1/4)*A
4A=-2sin(lnx)/x^2-cos(lnx)/x^2-A
A=(-2/5)*sin(lnx)/x^2-(1/5)*cos(lnx)/x^2+C，其中C是任意常数
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