遭難者氏の積分

$$\begin{array}{rl}{\int }_0^{\frac{\pi }{2}}\cos \ln \tan \theta d\theta & ={\int }_0^{\frac{\pi }{2}}\mathrm{Re}(e^{i\ln \tan \theta })d\theta \\ & =\mathrm{Re}{\int }_0^{\frac{\pi }{2}}{\tan }^i\theta d\theta \\ & =\mathrm{Re}{\int }_0^{\frac{\pi }{2}}{\sin }^i\theta {\cos }^{-i}\theta d\theta \\ & =\mathrm{Re}{\int }_0^{\frac{\pi }{2}}{\sin }^{2\frac{i+1}{2}-1}\theta {\cos }^{2\frac{-i+1}{2}-1}\theta d\theta \\ & =\mathrm{Re}\left\{\frac{1}{2}\mathrm{B}(\frac{i+1}{2},\frac{-i+1}{2})\right\}\\ & =\frac{1}{2}\mathrm{Re}\left\{\frac{\mathrm{\Gamma }\left(\frac{i+1}{2}\right)\mathrm{\Gamma }\left(\frac{-i+1}{2}\right)}{\mathrm{\Gamma }(1)}\right\}\\ & =\frac{1}{2}\mathrm{Re}\left\{\frac{\pi }{\sin (\pi \cdot \frac{i+1}{2})}\right\}\\ & =\frac{\pi }{2}\mathrm{Re}\left\{\frac{1}{\sin \frac{i\pi }{2}\cos \frac{\pi }{2}+\sin \frac{\pi }{2}\cos \frac{i\pi }{2}}\right\}\\ & =\frac{\pi }{2}\mathrm{Re}\left\{\frac{1}{\cos \frac{i\pi }{2}}\right\}\\ & =\frac{\pi }{2}\mathrm{Re}\left\{\frac{1}{\cosh \frac{\pi }{2}}\right\}\\ & =\frac{\pi }{2}\mathrm{Re}\left\{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{\pi }{2}\right\}\\ & =\frac{\pi }{2}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{\pi }{2}\end{array}$$