Solve Error in Integrating z^2lnz from 0 to 2

[MathewsMD]

Homework Statement

$\int_0^2 z^2lnzdz$


2. The attempt at a solution

$u = lnz, dz = zdu, lim_{t→-∞}\int_t^{ln2} ue^udu$
$lim_{t→-∞}ue^u - e^u l^{ln2}_t$
$2(ln2 - 1) - lim_{t→-∞}\frac {t-1}{e^{-t}}$ Using l'Hospital's rule:
$2(ln2 - 1)$

This is incorrect though. Any pointers on where I went wrong? I believe the domain tom -∞ to ln2 has no discontinuities for the newly defined function and don't seem to see any blatant errors...

All help is welcomed and greatly appreciated!

 

[jackarms]

Looks like an error in substituting. When you solve for dz, when substituting in you should get:

$\int^{2}_{0}z^{3}lnzdu$, then with $z = e^{u}$, it should become:

$\int^{ln2}_{-\infty}ue^{3u}du$

 

[Saitama]

I would suggest Integration by Parts rather than a substitution.