積分　x^2*√(x^2+1)

$3$\displaystyle{ \int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$

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

部分積分法を用いて計算をすすめる．

$3$\displaystyle{=\int x{ \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{ $ {x}^{2}+1 $ }^{ \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } \} }^{\prime }dx }$

$3$\displaystyle{=x\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{ $ {x}^{2}+1 $ }^{ \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} }}$ $3$\displaystyle{- \int {x}^{\prime }\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}{ $ {x}^{2}+1 $ }^{ \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} }dx }$

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

$3$\displaystyle{ \int \sqrt{ {x}^{2}+1 }dx }$ の積分はこの積分の $3$\displaystyle{a=1}$ の場合である．よって

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

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}x $ {x}^{2}+1 $ \sqrt{ {x}^{2}+1 }- \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 }}$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| }$

$3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 } \{ 2 $ {x}^{2}- 1 $ +1 \} }$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| }$

$3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 } $ 2{x}^{2}+1 $ }$ $3$\displaystyle{- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| }$

このように，左辺と同じ積分 $3$\displaystyle{\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ が右辺にも現れた．

よって，上記の方程式を $3$\displaystyle{\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ について解く．

$3$\displaystyle{\int {x}^{2}\sqrt{ {x}^{2}+1 }dx +\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 } $ 2{x}^{2}+1 $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| }$

$3$\displaystyle{\frac{\hspace{2}4\hspace{2}}{\hspace{2}3\hspace{2}}\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 } $ 2{x}^{2}+1 $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| }$

$3$\displaystyle{\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{=\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}} \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}x\sqrt{ {x}^{2}+1 } $ 2{x}^{2}+1 $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\log \| x+\sqrt{ {x}^{2}+1 } \| \} }$

$3$\displaystyle{\int {x}^{2}\sqrt{ {x}^{2}+1 }dx }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}8\hspace{2}} \{ x\sqrt{ {x}^{2}+1 } $ 2{x}^{2}+1 $ - \log \| x+\sqrt{ {x}^{2}+1 } \| \} }$

最後に，積分定数 $3$\displaystyle{C}$ を付け加えると

となる．

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最終更新日： 2023年10月4日

積分 x^2*√(x^2+1)

積分　x^2*√(x^2+1)