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

1 Answer
Apr 28, 2018

$= - \left(\frac{2 {\left(\ln x\right)}^{2} + 2 \left(\ln x\right) + 1}{4 {x}^{2}}\right) + \text{ const}$

Explanation:

Starting with:

$\int q \quad {\left(\ln x\right)}^{2} / {x}^{3} \setminus \mathrm{dx}$

$z = \ln x , q \quad x = {e}^{z} , q \quad \mathrm{dx} = {e}^{z} \setminus \mathrm{dz}$

$= \setminus \int \setminus q \quad {z}^{2} / \left({e}^{3 z}\right) {e}^{z} \setminus \mathrm{dz}$

$= \setminus \int \setminus q \quad {z}^{2} \setminus {e}^{- 2 z} \setminus \mathrm{dz}$

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

$= - \left(\frac{2 {\left(\ln x\right)}^{2} + 2 \left(\ln x\right) + 1}{4 {x}^{2}}\right) + \text{ const}$