∫ 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 ) = ∫ 0 π 4 1 cos 2 ( θ ) + 1 = tan ( θ ) + θ | 0 π 4 = tan ( π 4 ) + π 4 − ( tan ( θ ) + 0 ) = 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}}{\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}}}
