$\underset{-1}{\overset{3}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}+Ta{n}^{-1}\frac{{\displaystyle x^2+1}}{{\displaystyle x}}\right)dx=$

$$\begin{gathered} \underset{-1}{\overset{3}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}+Ta{n}^{-1}\frac{{\displaystyle x^2+1}}{{\displaystyle x}}\right)dx \\ =\underset{-1}{\overset{0}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}+Ta{n}^{-1}\left(\frac{{\displaystyle x^2+1}}{{\displaystyle x}}\right)\right) \\ +\underset{0}{\overset{3}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}+Ta{n}^{-1}\frac{{\displaystyle x^2+1}}{{\displaystyle x}}\right)dx \\ =\underset{-1}{\overset{0}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}-\pi +co{t}^{-1}\left(\frac{x}{x^2+1}\right)\right)dx \\ +\underset{0}{\overset{3}{\int }}\left(Ta{n}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}+co{t}^{-1}\frac{{\displaystyle x}}{{\displaystyle x^2+1}}\right)dx \\ \left({\tan }^{-1}x=co{t}^{-1}\frac{1}{x}if x>0\right) \\ =-\pi +co{t}^{-1}\frac{1}{x}, x<0 \\ =\underset{-1}{\overset{0}{\int }}\left(\frac{\pi }{2}-\pi \right)+\underset{0}{\overset{3}{\int }}\frac{\pi }{2}dx \\ ={\left(-\frac{\pi }{2}x\right)}_{-1}^0+{\left(\frac{\pi }{2}x\right)}_0^3 \\ =\left(-\frac{\pi }{2}\right)\left(0-(-1)\right)+\frac{\pi }{2}\left(3-0\right) \\ =-\frac{\pi }{2}+\frac{3\pi }{2}=\pi \end{gathered}$$