定义

Sn=∫0π/2sin⁡nxdx S_n = \int_0^{\pi/2} \sin^n x dx

结论

Sn=n−1nSn−2S0=π2S1=1 \begin{aligned} & S_n = \frac{n-1}{n}S_{n-2} \\ & S_0 = \frac{\pi}{2} \\ & S_1 = 1 \end{aligned}

另外注意

Sn=∫0π/2sin⁡nxdx=∫0π/2cos⁡nxdx S_n = \int_0^{\pi/2} \sin^n x dx = \int_0^{\pi/2} \cos^n x dx

推导

主要使用分部积分。S0S_0和S1S_1的计算这里省略。

另外可以注意，改变SnS_n的定义为余弦的积分，结论依然成立。

Sn=∫0π/2sin⁡nxdx=−∫0π/2sin⁡n−1xdcos⁡x=−[sin⁡n−1xcos⁡x∣0π/2−∫0π/2cos⁡xdsin⁡n−1x]=∫0π/2cos⁡2x(n−1)sin⁡n−2xdx=∫0π/2(1−sin⁡2x)(n−1)sin⁡n−2xdx=(n−1)∫0π/2sin⁡n−2xdx−(n−1)∫0π/2sin⁡nxdx=(n−1)Sn−2−(n−1)Sn⇒Sn=n−1nSn−2 \begin{aligned} S_n &= \int_0^{\pi/2} \sin^n x dx \\ = & -\int_0^{\pi/2} \sin^{n-1} x d\cos x \\ = & -\left[\left. \sin^{n-1} x\cos x\right|_0^{\pi/2} - \int_0^{\pi/2}\cos x d\sin^{n-1} x\right] \\ = & \int_0^{\pi/2}\cos^2 x (n-1)\sin^{n-2} x dx \\ = & \int_0^{\pi/2}(1-\sin^2 x)(n-1)\sin^{n-2} x dx \\ = & (n-1)\int_0^{\pi/2}\sin^{n-2}xdx - (n-1)\int_0^{\pi/2} \sin^n xdx \\ = & (n-1)S_{n-2} - (n-1)S_n \\ \Rightarrow & S_n = \frac{n-1}{n}S_{n-2} \end{aligned}