How do I solve $\int_1^{2}\frac{(\ln{x})^2}{x^3} dx$? $\int_1^{2}\frac{(\ln{x})^2}{x^3} dx$
This seemed like an integration by parts problem, so I used it and got:
$\int_1^{2}\frac{(\ln{x})^2}{x^3} dx = [-\frac{1}{2}x^{-2}(\ln{x})^2 | _{1}^{2}] - \int_1^{2}\frac{2\ln{x}}{x^3}dx$
But I can't get anywhere with the $\int_1^{2}\frac{2\ln{x}}{x^3}dx$ part, making me think I didn't choose the right approach. Am I missing something simple?

Hint :
Put $\ln x=t$ and hence $dx=e^t dt$
The integral then changes to $$\int_{0}^{\ln 2} t^2e^{-2t}dt$$
Now apply repeated integration by parts with $u=t^2$ and $dv=e^{-2t}dt$
