5.4 Indefinite Integrals and the Net Change Theorem/37

${\begin{aligned}\int _{0}^{\frac {\pi }{4}}\left({\frac {1+\cos ^{2}(\theta )}{\cos ^{2}(\theta )}}\right)d\theta &=\int _{0}^{\frac {\pi }{4}}\left({\frac {1}{\cos ^{2}(\theta )}}+{\frac {\cos ^{2}(\theta )}{\cos ^{2}(\theta )}}\right)d\theta &=\tan({\theta })+\theta \ {\bigg |}_{0}^{\frac {\pi }{4}}\\[2ex]&=\tan({\frac {\pi }{4}})+{\frac {\pi }{4}}\\[2ex]&=1+{\frac {\pi }{4}}\end{aligned}}$

$=\int _{0}^{\frac {\pi }{4}}\left(\sec ^{2}(\theta )+1\right)d\theta$