ウォリス積分

$3$ \displaystyle{\int }_{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }{ \cos }^{n}xdx$ $3$\displaystyle{ =\displaystyle{\int }_{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }{ \sin }^{n}xdx }$

$3$\displaystyle{=\left{{ \begin{array} \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2} n- 3 \hspace{2}}{\hspace{2} n- 2 \hspace{2}}\cdots \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} & \vspace{6}\\ \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2} n- 3 \hspace{2}}{\hspace{2} n- 2 \hspace{2}}\cdots \frac{\hspace{2}2\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1 & \vspace{6}\\ \end{array} }$

$3$n$ ：偶数

$3$n$ ：奇数

この積分は，下図の赤の領域の面積を求めていることになる．
積分区間が0から $3$\displaystyle{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }$ でない場合は，周期性，対称性に注意して適用すること．

$3$\displaystyle{{\sin }^{2}x}$ のグラフ（青線： $3$\displaystyle{\sin x}$ ）	$3$\displaystyle{{\cos }^{2}x}$ のグラフ（青線： $3$\displaystyle{\cos x}$ ）
$3$\displaystyle{{\sin }^{3}x}$ のグラフ（青線： $3$\displaystyle{\sin x}$ ）	$3$\displaystyle{{\cos }^{3}x}$ のグラフ（青線： $3$\displaystyle{\cos x}$ ）
$3$\displaystyle{{\sin }^{4}x}$ のグラフ（青線： $3$\displaystyle{\sin x}$ ）	$3$\displaystyle{{\cos }^{4}x}$ のグラフ（青線： $3$\displaystyle{\cos x}$ ）

■計算例

$3$\displaystyle{\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }{\sin }^{5}xdx}$ $3$\displaystyle{=\frac{\hspace{2}4\hspace{2}}{\hspace{2}5\hspace{2}}\cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1}$ $3$\displaystyle{=\frac{\hspace{2}8\hspace{2}}{\hspace{2} 15 \hspace{2}}}$

$3$\displaystyle{ \displaystyle{\int }_{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }{ \sin }^{6}xdx }$ $3$\displaystyle{ =\frac{\hspace{2}5\hspace{2}}{\hspace{2}6\hspace{2}}\cdot \frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 5{\pi} \hspace{2}}{\hspace{2} 32 \hspace{2}} }$

$3$\displaystyle{ \int _{0}^{ 2{\pi} } { \cos }^{4}xdx }$ $3$\displaystyle{ =4\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \cos }^{4}xdx }$ $3$\displaystyle{ =4\cdot \frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}=\frac{\hspace{2} 3{\pi} \hspace{2}}{\hspace{2}4\hspace{2}} }$

$3$\displaystyle{ \int _{ - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \cos }^{3}xdx }$ $3$\displaystyle{ =2\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \cos }^{3}xdx=2 \cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1=\frac{\hspace{2}4\hspace{2}}{\hspace{2}3\hspace{2}} }$

$3$\displaystyle{ \int _{0}^{{\pi}} { \sin }^{3}xdx }$ $3$\displaystyle{ =2\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \sin }^{3}xdx=2 \cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1=\frac{\hspace{2}4\hspace{2}}{\hspace{2}3\hspace{2}} }$

■ $3$\displaystyle{ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \sin }^{n}xdx }$ の計算

●解説動画

◇積分の動画一覧のページへ

●式の導出

$3$\displaystyle{ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \sin }^{n}xdx =\hspace{1}{I}_{n} }$

とおく．部分積分によって

$3$\displaystyle{ \begin{array}{lll} \hspace{1}{I}_{n} & =\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } $ \sin x $ $ { \sin }^{ n- 1 }x $ dx & \vspace{6}\\ & = \[ $ - \cos x $ $ { \sin }^{ n- 1 }x $ \] _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }- \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } $ - \cos x $ $ n- 1 $ $ { \sin }^{ n- 2 }x $ \cos xdx & \vspace{6}\\ & = $ n- 1 $ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } $ { \cos }^{2}x $ $ { \sin }^{ n- 2 }x $ dx & \vspace{6}\\ & = $ n- 1 $ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } $ 1- { \sin }^{2}x $ $ { \sin }^{ n- 2 }x $ dx & \vspace{6}\\ & = $ n- 1 $ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } $ { \sin }^{ n- 2 }x $ dx - $ n- 1 $ \int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } { \sin }^{n}xdx & \vspace{6}\\ & = $ n- 1 $ \hspace{1}{I}_{ n- 2 }- $ n- 1 $ \hspace{1}{I}_{n} & \vspace{6}\\ \end{array} }$

となる．よって，漸化式

$3$\displaystyle{\begin{array}{lll}\hspace{1}{I}_{n}= $ n- 1 $ \hspace{1}{I}_{ n- 2 }- $ n- 1 $ \hspace{1}{I}_{n}& \vspace{6}\\ \hspace{1}{I}_{n}=\frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\hspace{1}{I}_{ n- 2 }& \vspace{6}\\ \end{array}}$

が得られる．

$3$\displaystyle{ \hspace{1}{I}_{0}=\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } dx =\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }$ ， $3$\displaystyle{ \hspace{1}{I}_{1}=\int _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} } \sin xdx = \[ - \cos x \] _{0}^{ \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}} }=1 }$

であるので

$3$\displaystyle{\hspace{1}{I}_{n}=\frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2} n- 3 \hspace{2}}{\hspace{2} n- 2 \hspace{2}}\cdot \cdots \cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}}$ （ $3$n$ が偶数の場合）

$3$\displaystyle{\hspace{1}{I}_{n}=\frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2} n- 3 \hspace{2}}{\hspace{2} n- 2 \hspace{2}}\cdot \cdots \cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2}3\hspace{2}}\cdot 1}$ （ $3$n$ が奇数の場合）

となる．