積分　√(x^2-a^2)

$3$\displaystyle{ \int \sqrt{ {x}^{2}-{a}^{2} }dx }$ $3$\displaystyle{ $ a> 0 $ }$

部分積分法より． $3$\displaystyle{{f}^{\prime } $x$ =1,g $x$ =\sqrt{ {x}^{2}-{a}^{2} }}$ とおく．

$3$\displaystyle{=\int 1\cdot \sqrt{ {x}^{2}-{a}^{2} }dx }$

$3$\displaystyle{=x\sqrt{ {x}^{2}-{a}^{2} }- \int \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} \sqrt{ {x}^{2}-{a}^{2} } \hspace{2}}dx }$

$3$\displaystyle{=x\sqrt{ {x}^{2}-{a}^{2} }- \int \frac{\hspace{2} {x}^{2}-{a}^{2}+{a}^{2} \hspace{2}}{\hspace{2} \sqrt{ {x}^{2}-{a}^{2} } \hspace{2}}dx }$

$3$\displaystyle{=x\sqrt{ {x}^{2}-{a}^{2} }}$ $3$\displaystyle{- \int \sqrt{ {x}^{2}-{a}^{2} }dx -\int \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} \sqrt{ {x}^{2}-{a}^{2} } \hspace{2}}dx }$

$3$\displaystyle{=x\sqrt{ {x}^{2}-{a}^{2} }}$ $3$\displaystyle{- \int \sqrt{ {x}^{2}-{a}^{2} }dx -{a}^{2}\log \| x+\sqrt{ {x}^{2}-{a}^{2} } \| }$

$3$\displaystyle{\int \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} \sqrt{ {x}^{2}-{a}^{2} } \hspace{2}}dx }$ の積分はここを参照のこと

改めて書き直すと

$3$\displaystyle{\displaystyle{ \int \sqrt{ {x}^{2}- {a}^{2} } }dx}$ $3$\displaystyle{=x\sqrt{ {x}^{2}- {a}^{2} }}$ $3$\displaystyle{- \displaystyle{ \int \sqrt{ {x}^{2}- {a}^{2} } }dx- {a}^{2}\log \| x+\sqrt{ {x}^{2}- {a}^{2} } \| }$

$3$\displaystyle{\int \sqrt{ {x}^{2}-{a}^{2} }dx }$ について整理すると

$3$\displaystyle{ 2\displaystyle{ \int \sqrt{ {x}^{2}- {a}^{2} } }dx }$ $3$\displaystyle{ =x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \| x+\sqrt{ {x}^{2}- {a}^{2} } \| }$

$3$\displaystyle{ \displaystyle{ \int \sqrt{ {x}^{2}- {a}^{2} } }dx }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \| x+\sqrt{ {x}^{2}- {a}^{2} } \| \} }$

積分定数を付け加えると

$3$\displaystyle{\displaystyle{ \int \sqrt{ {x}^{2}- {a}^{2} } }dx}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \| x+\sqrt{ {x}^{2}- {a}^{2} } \| \} +C}$

●別解

$3$\displaystyle{x=\frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}}}$ とおく（置換積分）．ただし， $3$\displaystyle{0< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}}$ ．

$3$\displaystyle{\frac{\hspace{2} dx \hspace{2}}{\hspace{2} d{\theta} \hspace{2}}=\frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}\rightarrow dx=\frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta}}$

よって

与式 $3$\displaystyle{=\displaystyle{ \int \sqrt{ { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}} }\right) }^{2}- {a}^{2} }\cdot }\frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta}}$

$3$\displaystyle{=\displaystyle{ \int a\sqrt{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}- 1 }\cdot }\frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta}}$

$3$\displaystyle{{\tan }^{2}{\theta}+1=\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}}$ より， $3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}- 1={\tan }^{2}{\theta}}$ ．よって

$3$\displaystyle{=\displaystyle{ \int a\sqrt{ { \tan }^{2}{\theta} }\cdot \frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta} }}$

$3$\displaystyle{0< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}}$ のとき， $3$\displaystyle{\tan {\theta}> 0}$ となり， $3$\displaystyle{\sqrt{ { \tan }^{2}{\theta} }=\tan {\theta}}$ ．

$3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}}$ のとき， $3$\displaystyle{\tan {\theta}< 0}$ となり， $3$\displaystyle{\sqrt{ { \tan }^{2}{\theta} }=- \tan {\theta}}$ ．

よって，場合分けをして計算を進める．

I． $3$\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}$ の場合

与式 $3$\displaystyle{=\displaystyle{ \int a\tan {\theta}\cdot \frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta} }}$

$3$\displaystyle{={a}^{2}\displaystyle{ \int \frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2} { \cos }^{3}{\theta} \hspace{2}}d{\theta} }}$

$3$\displaystyle{={a}^{2}\displaystyle{ \int \frac{\hspace{2} 1- { \cos }^{2}{\theta} \hspace{2}}{\hspace{2} { \cos }^{3}{\theta} \hspace{2}}d{\theta} }}$

$3$\displaystyle{={a}^{2}\left({ \displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{3}{\theta} \hspace{2}}d{\theta} }- \displaystyle{ \int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}}d{\theta} } }\right)}$

$3$\displaystyle{\int \frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{3}x \hspace{2}}dx}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2} \sin x \hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\log \left({ \frac{\hspace{2} 1+\sin x \hspace{2}}{\hspace{2} 1- \sin x \hspace{2}} }\right)+C}$

$3$\displaystyle{\int \frac{\hspace{2}1\hspace{2}}{\hspace{2} \cos x \hspace{2}}dx}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} 1+\sin x \hspace{2}}{\hspace{2} 1- \sin x \hspace{2}} }\right)+C}$ より

$3$\displaystyle{=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} \sin x \hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} 1+\sin x \hspace{2}}{\hspace{2} 1- \sin x \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{x=\frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}}}$ より， $3$\displaystyle{\cos {\theta}=\frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}}}$

$3$\displaystyle{\sin {\theta}=\sqrt{ 1- { \cos }^{2}{\theta} }}$ $3$\displaystyle{=\sqrt{ 1- { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} }}$ $3$\displaystyle{\displaystyle{=\sqrt{ 1- { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} }}}$ $3$\displaystyle{=\frac{\hspace{2} \sqrt{ 1- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}}}$ ．

（∵ $3$\displaystyle{0< {\theta}< \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}}$ のとき， $3$\displaystyle{\cos {\theta}> 0}$ ，よって， $3$\displaystyle{x> 0}$ ，したがって， $3$\displaystyle{\sqrt{ {x}^{2} }=x}$ ）

よって

$3$\displaystyle{=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} \frac{\hspace{2} \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}}{\hspace{2} { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} 1+\frac{\hspace{2} \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}}{\hspace{2} 1- \frac{\hspace{2} \sqrt{ 1{x}^{2}{a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} x+\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} x- \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} { \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }^{2} \hspace{2}}{\hspace{2} \left({ x- \sqrt{ {x}^{2}- {a}^{2} } }\right)\left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} { \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right)- {a}^{2}\log a }\right}+C}$

$3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{a}^{2}\log a+C}$ を改めて $3$\displaystyle{C}$ とおくと

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }\right}+C}$

II． $3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}}$ の場合

与式 $3$\displaystyle{=\displaystyle{ \int a\left({ - \tan {\theta} }\right)\cdot \frac{\hspace{2} a\sin {\theta} \hspace{2}}{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}d{\theta} }}$

$3$\displaystyle{=- {a}^{2}\displaystyle{ \int \frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2} { \cos }^{3}{\theta} \hspace{2}}d{\theta} }}$

上記の計算を参考にすると

$3$\displaystyle{=- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} \sin x \hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} 1+\sin x \hspace{2}}{\hspace{2} 1- \sin x \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{x=\frac{\hspace{2}a\hspace{2}}{\hspace{2} \cos {\theta} \hspace{2}}}$ より， $3$\displaystyle{\cos {\theta}=\frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}}}$

$3$\displaystyle{\sin {\theta}=\sqrt{ 1- { \cos }^{2}{\theta} }}$ $3$\displaystyle{=\sqrt{ 1- { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} }}$ $3$\displaystyle{\displaystyle{=\sqrt{ 1- { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} }}}$ $3$\displaystyle{=- \frac{\hspace{2} \sqrt{ 1- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}}}$ ．

（∵ $3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}}$ のとき， $3$\displaystyle{\cos {\theta}< 0}$ ，よって， $3$\displaystyle{x< 0}$ ，したがって， $3$\displaystyle{\sqrt{ {x}^{2} }=- x}$ ）

よって

$3$\displaystyle{=- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \frac{\hspace{2} - \frac{\hspace{2} \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}}{\hspace{2} { \left({ \frac{\hspace{2}a\hspace{2}}{\hspace{2}x\hspace{2}} }\right) }^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} 1- \frac{\hspace{2} \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}}{\hspace{2} 1+\frac{\hspace{2} \sqrt{ 1{x}^{2}{a}^{2} } \hspace{2}}{\hspace{2}x\hspace{2}} \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ - \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} x- \sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} x+\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ - \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right)\left({ x- \sqrt{ {x}^{2}- {a}^{2} } }\right) \hspace{2}}{\hspace{2} { \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }^{2} \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2}2\hspace{2}}\left{{ - \frac{\hspace{2} x\sqrt{ {x}^{2}- {a}^{2} } \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\log \left({ \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} { \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }^{2} \hspace{2}} }\right) }\right}+C}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right)+{a}^{2}\log a }\right}+C}$

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{a}^{2}\log a+C}$ を改めて $3$\displaystyle{C}$ とおくと

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }\right}+C}$

I，IIより

$3$\displaystyle{ \int \sqrt{ {x}^{2}-{a}^{2} }dx }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ x\sqrt{ {x}^{2}- {a}^{2} }- {a}^{2}\log \left({ x+\sqrt{ {x}^{2}- {a}^{2} } }\right) }\right}+C}$

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最終更新日： 2025年9月9日

積分 √(x^2-a^2)

●別解

I．の場合

II． の場合

I，IIより

積分　√(x^2-a^2)

I． $3$\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}$ の場合

II． $3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}< {\theta}< {\pi}}$ の場合