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

1 Answer
Apr 28, 2018

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

Explanation:

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"