Question Number 197384 by Erico last updated on 15/Sep/23
![If f(x)=∫^( x) _( (1/x)) ((lnt)/(t^2 −1))arctan(t)dt Prove that: • ∀x>0 f(x)= (π/8) ∫^( π) _( 0) arctan[(1/2)(x−(1/x))sint]dt •lim_(x→+∞) f(x)=(π^3 /(16))](https://www.tinkutara.com/question/Q197384.png)
$$\mathrm{If}\:{f}\left({x}\right)=\underset{\:\frac{\mathrm{1}}{\mathrm{x}}} {\int}^{\:\:\mathrm{x}} \frac{{lnt}}{{t}^{\mathrm{2}} −\mathrm{1}}{arctan}\left({t}\right){dt} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\bullet\:\forall{x}>\mathrm{0}\:\:\:\:\:\:\:\:{f}\left({x}\right)=\:\frac{\pi}{\mathrm{8}}\:\underset{\:\mathrm{0}} {\int}^{\:\pi} \mathrm{arctan}\left[\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{sint}\right]\mathrm{dt} \\ $$$$\bullet\underset{\mathrm{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right)=\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$