几个反正切函数复合三角函数的定积分

青青子衿

\begin{align*}
\int_0^{\frac{\pi}{2}}\arctan\left(\cos^2x\right)\mathrm{d}x\,\,&\dot{=}\,\,\int_0^{\frac{\pi}{2}}\arctan\left(\sin^2x\right)\mathrm{d}x=\frac{\pi}{2}\arctan\sqrt{\frac{\sqrt{2}-1}{2}}\\
\int_0^{\frac{\pi}{2}}\arctan\left(\sec^2x\right)\mathrm{d}x\,\,&\dot{=}\,\,\int_0^{\frac{\pi}{2}}\arctan\left(\csc^2x\right)\mathrm{d}x=\frac{\pi^2}{4}-\frac{\pi}{2}\arctan\sqrt{\frac{\sqrt{2}-1}{2}}\\
\int_0^{\frac{\pi}{2}}\arctan\left(\tan^2x\right)\mathrm{d}x\,\,&\dot{=}\,\,\int_0^{\frac{\pi}{2}}\arctan\left(\cot^2x\right)\mathrm{d}x=\frac{\pi^2}{8}\\
\end{align*}