∫ 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 = [ tan ( π 4 ) + π 4 ] − [ tan 0 + 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}}\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]&=\left[\tan \left({\frac {\pi }{4}}\right)+{\frac {\pi }{4}}\right]-\left[\tan {0}+0\right]\\[2ex]&=1+{\frac {\pi }{4}}\end{aligned}}}
![{\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]&=\left[\tan \left({\frac {\pi }{4}}\right)+{\frac {\pi }{4}}\right]-\left[\tan {0}+0\right]\\[2ex]&=1+{\frac {\pi }{4}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/413cd7417c6d923fea9097c2cd247f4aa2d96f38)