求$\int_0^{\frac {\pi}4}\cos^82x\mathrm dx$
源自知乎提问
题:求 $\int_0^{\frac {\pi}4}\cos^82x\mathrm dx$.
直接算算看,次数不算高.
明显的 $\int_0^{\frac {\pi}4}\cos^82x\mathrm dx=\frac 12\int_0^{\frac {\pi}2}\cos^8t\mathrm dt,$ 下面先考察
$$\int_0^{\frac {\pi}2}\cos^8x\mathrm dx.$$
由分部积分法
\begin{align*} \end{align*}$$\begin{align*} \int_0^{\frac {\pi}2}\cos^8x\mathrm dx&=\int_0^{\frac {\pi}2}\cos^7x\mathrm d(\sin x)\\[1em] &=\cos^7x\sin x\bigg|_{0}^{\pi/2}-\int_0^{\frac {\pi}2}\sin x\mathrm d(\cos^7x)\\[1em] &=0+7\int_0^{\frac {\pi}2}\cos^6 x\sin^2 x\mathrm dx\\[1em] &=7\int_0^{\frac {\pi}2}\cos^6 x(1-\cos^2 x)\mathrm dx\\[1em] &=7\int_0^{\frac {\pi}2}\cos^6 x-7\int_0^{\frac {\pi}2}\cos^8 x\mathrm dx\\[1em] \iff 8\int_0^{\frac {\pi}2}\cos^8x\mathrm dx&=7\int_0^{\frac {\pi}2}\cos^6 x\mathrm dx\\[1em] \color{red}{\int_0^{\frac {\pi}2}\cos^8x\mathrm dx}\ &\color{red}{=\frac 78\int_0^{\frac {\pi}2}\cos^6 x\mathrm dx}, \end{align*}
对 $\int_0^{\frac {\pi}2}\cos^6 x\mathrm dx$ 重复这一过程,进行迭代有 $\int_0^{\frac {\pi}2}\cos^6 x\mathrm dx=\frac 56\int_0^{\frac {\pi}2}\cos^4 x\mathrm dx,$ 再次迭代两次,则
\begin{align*} \color{red}{\int_0^{\frac {\pi}2}\cos^8x\mathrm dx}\ &=\frac 78\cdot \frac 56\cdot \frac 34\cdot \frac 12\int_0^{\frac {\pi}2}\mathrm dx\\[1em] &=\frac {7!!}{8!!}\cdot \frac {\pi}2. \end{align*}
从而 $\int_0^{\frac {\pi}4}\cos^82x\mathrm dx=\frac {105\pi}{1536}.$
PS:真的忘记了,这么好玩的积分~. |