5.4 Indefinite Integrals and the Net Change Theorem/37

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

&= \left[\tan\left({\frac{\pi}{4}}\right) + \frac{\pi}{4}\right]

- left[\tan{0} + 0\right] \\[2ex]