6.1 Areas Between Curves/25

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \color{red}\mathbf{y=x^{2}} & \color{royalblue}\mathbf{y=\frac{2}{({x^2}+1)}} \\ & x=-1 & x=1 \end{align} }

${\begin{aligned}\int _{1}^{-1}\left({\frac {2}{{x^{2}}+1}}-x^{2}\right)dx&=\int _{1}^{-1}\left(2\cdot {\frac {1}{x^{2}+1}}\right)-\left(x^{2}\right)dx\\[2ex]&=\left[2\arctan(x)-{\frac {x^{3}}{3}}\right]{\Bigg |}_{-1}^{1}\\[2ex]&=\left[2\arctan(1)-{\frac {(1)^{3}}{3}}\right]-\left[2\arctan(-1)-{\frac {(-1)^{3}}{3}}\right]\\[2ex]&=\left[{\frac {2\pi }{4}}-{\frac {1}{3}}\right]-\left[-{\frac {2\pi }{4}}-\left(-{\frac {1}{3}}\right)\right]\\[2ex]&={\frac {\pi }{2}}-{\frac {1}{3}}+{\frac {\pi }{2}}-{\frac {1}{3}}\\[2ex]&=\pi -{\frac {2}{3}}\end{aligned}}$