5.5 The Substitution Rule/55

$\int _{0}^{\pi }\sec ^{2}\left({\frac {t}{4}}\right)dt$

${\begin{aligned}u&={\frac {t}{4}}\\[2ex]du&={\frac {1}{4}}dt\\[2ex]4du&=dx\end{aligned}}$

${\begin{aligned}\int _{0}^{\pi }\sec ^{2}\left({\frac {t}{4}}\right)dt&=4\int _{0}^{\pi }\sec ^{2}(u)du\\[2ex]&=4\cdot \tan ^{2}(u)=4\cdot \tan ^{2}\left({\frac {t}{4}}\right){\bigg |}_{0}^{\pi }\\[2ex]&=4\cdot \tan ^{2}\left({\frac {\pi }{4}}\right)-4\cdot \tan ^{2}\left({\frac {0}{4}}\right)\\[2ex]&=4-0\\[2ex]&=4\end{aligned}}$