5.4 Indefinite Integrals and the Net Change Theorem/41

${\begin{aligned}\int _{0}^{\frac {1}{\sqrt {3}}}{\frac {t^{2}-1}{t^{4}-1}}dt&=\int _{0}^{\frac {1}{\sqrt {3}}}{\frac {(t^{2}-1)}{(t^{2}-1)(t^{2}+1)}}dt=\int _{0}^{\frac {1}{\sqrt {3}}}{\frac {1}{(t^{2}+1)}}dt\\[2ex]&=\arctan {(t)}{\bigg |}_{0}^{\frac {1}{\sqrt {3}}}\\[2ex]&=\arctan \left({\frac {1}{\sqrt {3}}}\right)-\arctan(0)\\[2ex]&={\frac {\pi }{6}}-0\\[2ex]&={\frac {\pi }{6}}\end{aligned}}$