一般化

\begin{align} \int_0^\frac{\pi}{2}\tan^z \theta \ d \theta &= \int_0^\frac{\pi}{2}\sin^z \theta \cos^{-z} \theta \ d \theta \\ &= \int_0^\frac{\pi}{2}\sin^{2\frac{z+1}{2}-1} \theta \cos^{2\frac{-z+1}{2}-1} \theta \ d \theta \\ &= \frac12 {\rm{B}}\left(\frac{z+1}{2}, \frac{-z+1}{2}\right) \\ &= \frac12 \frac{1}{\Gamma(1)}\Gamma\left(\frac{z+1}{2}\right)\Gamma\left(\frac{-z+1}{2}\right) \\ &= \frac{1}{2} \frac{\pi}{\sin\left(\pi\cdot\frac{z+1}{2}\right)} \\ &= \frac{\pi}{2} \frac{1}{\sin\frac{z\pi}{2}\cos\frac{\pi}{2}+\sin\frac{\pi}{2}\cos\frac{z\pi}{2}} \\ &= \frac{\pi}{2} \sec \frac{z\pi}{2} \end{align}