[tex]\displaystyle\int_{\frac{1}{\sqrt{3}}}^{1}~tan^{-1}\left( \cfrac{1}{x} \right)dx \\\\[-0.35em] ~\dotfill\\\\ u=tan^{-1}\left( \cfrac{1}{x} \right)\implies \cfrac{du}{dx}=-\cfrac{1}{1+x^2}~\hfill v=\displaystyle\int~1\cdot dx\implies v=x \\\\[-0.35em] ~\dotfill\\\\ xtan^{-1}\left( \cfrac{1}{x} \right)+\displaystyle\int ~ \cfrac{1}{1+x^2}\cdot x\qquad \leftarrow \textit{let's do integration by parts here} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]u_1=x\implies \cfrac{du_1}{dx}=1~\hfill v_1=\displaystyle\int~\cfrac{1}{1+x^2}\implies v_1=tan^{-1}(x) \\\\\\ xtan^{-1}(x)-\displaystyle\int~tan^{-1}(x)\implies \underline{xtan^{-1}(x)-xtan^{-1}(x)}-\ln(\sqrt{1+x^2}) \\\\\\ \ln(\sqrt{1+x^2})\qquad \leftarrow \textit{now let's put it together with the left-side part} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \left. xtan^{-1}\left( \cfrac{1}{x} \right) \right]_{\frac{1}{\sqrt{3}}}^{1} ~~-~~\left.\cfrac{}{} \ln(\sqrt{1+x^2}) \right]_{\frac{1}{\sqrt{3}}}^{1}[/tex]
[tex]~\dotfill\\\\ \stackrel{\textit{let's do firstly the left-hand-side}}{[1\cdot tan^{-1}(1)]-\cfrac{1}{\sqrt{3}}\cdot tan\left( \cfrac{~~ 1~~}{\frac{1}{\sqrt{3}}} \right)\implies} tan^{-1}(1)-\cfrac{1}{\sqrt{3}}\cdot tan^{-1}(\sqrt{3}) \\\\\\ \cfrac{\pi }{4}-\cfrac{1}{\sqrt{3}}\cdot \cfrac{\pi }{3}\implies \cfrac{\pi }{4}-\cfrac{\pi }{3\sqrt{3}}\implies \cfrac{\pi }{4}-\cfrac{\pi }{3\sqrt{3}}\cdot \cfrac{\sqrt{3}}{\sqrt{3}}\implies \boxed{\cfrac{\pi }{4}-\cfrac{\pi \sqrt{3}}{9}}[/tex]
[tex]\stackrel{\textit{now let's do the right-hand-side}}{\ln(\sqrt{1+1})-\ln\left( \sqrt{1+\frac{1}{3}} \right)\implies} \ln(\sqrt{2})-\ln\left( \sqrt{\frac{4}{3}} \right) \stackrel{\textit{let's apply the (-1) in front}}{\implies -\ln(\sqrt{2})+\ln\left( \sqrt{\frac{4}{3}} \right)} \\\\\\ \ln\left( \sqrt{\frac{4}{3}} \right)-\ln(\sqrt{2})\implies \ln\left( \frac{2}{\sqrt{3}} \right)-\ln(\sqrt{2})\implies \ln\left( \cfrac{~~ \frac{2}{\sqrt{3}}~~}{\sqrt{2}} \right)[/tex]
[tex]\ln\left( \cfrac{2\sqrt{2}}{\sqrt{3}} \right)\implies \ln\left( \cfrac{\sqrt{8}}{\sqrt{3}} \right)\implies \ln\left( \sqrt{\cfrac{8}{3}} \right)\implies \ln\left[ \left( \cfrac{8}{3} \right)^{\frac{1}{2}} \right]\implies \boxed{\cfrac{1}{2}\ln\left( \cfrac{8}{3} \right)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\pi }{4}-\cfrac{\pi \sqrt{3}}{9}~~ + ~~\cfrac{1}{2}\ln\left( \cfrac{8}{3} \right)\implies \boxed{\left( \cfrac{1}{4}-\cfrac{\sqrt{3}}{9} \right)\pi ~~ + ~~\cfrac{1}{2}\ln\left( \cfrac{8}{3} \right)}[/tex]
