積分解説02

2020/11/02に出題した問題です。
https://twitter.com/sounansya_29/status/1323227259084271616?s=21

[解説]
$\begin{array}{rcl}& & {\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{\cos x{\displaystyle \sqrt{\log \tan x}}}{{\sin }^{3}x}dx\\ & =& {\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{{\displaystyle (1+{\tan }^{2}x)\sqrt{\log \tan x}}}{{\tan }^{3}x}dx\\ & =& {\int }_0^{\mathrm{\infty }}\frac{(1+e^{2t})e^t\sqrt{t}}{e^{3t}(1+e^{2t})}dt& (t=\log \tan x)\\ & =& {\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}}{e}^{-2t}dt\\ & =& \frac{1}{2\sqrt{2}}{\int }_0^{\mathrm{\infty }}{u}^{\frac{1}{2}}{e}^{-u}du& (u=2t)\\ & =& \frac{1}{2\sqrt{2}}\mathrm{\Gamma }\left(\frac{3}{2}\right)\\ & =& \frac{\sqrt{\pi }}{4\sqrt{2}}\end{array}$
よって、この問題の解答は${\displaystyle \frac{\sqrt{\pi }}{4\sqrt{2}}}$となります。