What is #int (lnx)^2 / x^3#?

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Answer 1 · 2018-04-28T23:54:16

#= - ( (2 (ln x)^2 + 2 (ln x) + 1)/(4 x^2) ) + " const"#

Starting with:

#int qquad (lnx)^2 / x^3 \ dx#

#z = ln x, qquad x = e^z, qquad dx = e^z \ dz #

#= \ int \ qquad z^2 / (e^(3z) )e^z \ dz#

#= \ int \ qquad z^2 \ e^(-2z) \ dz#

That's a whole load of IBP, which is very mechanical, and leads to this when you work it out by hand:

#= - ( (2 (ln x)^2 + 2 (ln x) + 1)/(4 x^2) ) + " const"#