∫ 0 π sec 2 ( t 4 ) d t {\displaystyle \int _{0}^{\pi }\sec ^{2}\left({\frac {t}{4}}\right)dt}
u = t 4 d u = 1 4 d t 4 d u = d x {\displaystyle {\begin{aligned}u&={\frac {t}{4}}\\[2ex]du&={\frac {1}{4}}dt\\[2ex]4du&=dx\end{aligned}}}
∫ 0 π sec 2 ( t 4 ) d t = 4 ∫ 0 π sec 2 ( u ) d u = 4 ⋅ tan 2 ( u ) = 4 ⋅ tan 2 ( t 4 ) | 0 π = 4 ⋅ tan 2 ( π 4 ) − 4 ⋅ tan 2 ( 0 4 ) = 4 − 0 = 4 {\displaystyle {\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}}}
![{\displaystyle {\begin{aligned}u&={\frac {t}{4}}\\[2ex]du&={\frac {1}{4}}dt\\[2ex]4du&=dx\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8676fb10a6e47b14bf4106e3278668355276489f)
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c7cb75f2537ed72892fca04d3f211d060d87821a)