5.4 Indefinite Integrals and the Net Change Theorem/37

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