5.5 The Substitution Rule/55

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

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

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

= 4\cdot \tan^2(u) &= 4\int_{0}^{\pi} \sec^2(u)du \\[2ex]