Question Number 222418 by Jgrads last updated on 26/Jun/25
![Prove that: lim_(n→+∞) [ ln^2 (n)−2∫^( n) _( 0) ((lnt)/( (√(1+t^2 )))) dt ]= (π^2 /6)+ln^2 (2)](https://www.tinkutara.com/question/Q222418.png)
$$\mathrm{Prove}\:\mathrm{that}:\: \\ $$$$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\:\left[\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{n}\right)−\mathrm{2}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{n}} \frac{\mathrm{lnt}}{\:\sqrt{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\:\mathrm{dt}\:\right]=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}+\mathrm{ln}^{\mathrm{2}} \left(\mathrm{2}\right) \\ $$
Answered by MrGaster last updated on 26/Jun/25
