∫ 0 π 4 ( 1 + cos 2 ( θ ) cos 2 ( θ ) ) d θ = ∫ 0 π 4 ( 1 cos 2 ( θ ) + cos 2 ( θ ) cos 2 ( θ ) ) d θ = ∫ 0 π 4 ( sec 2 ( θ ) + 1 ) d θ = tan ( θ ) + θ | 0 π 4 = 1 + π 4 {\displaystyle {\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]&=\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]
![{\displaystyle {\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]&=\tan({\theta })+\theta \ {\bigg |}_{0}^{\frac {\pi }{4}}\\[2ex]&=1+{\frac {\pi }{4}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/16dbec9e4b3915dbf64f0bf3487833c005b8a375)