逆余割関数の積分計算

\begin{alignat}{2}
&(1) \displaystyle\int \csc^{-1} axdx=x \csc^{-1} ax+\frac{1}{a} \tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\\
&(2) \displaystyle\int x \csc^{-1} axdx=\frac{1}{2}x^2 \csc^{-1} ax+\frac{x}{2a}\sqrt{1-\frac{1}{a^2x^2}}\\
&(3) \displaystyle\int x^2 \csc^{-1} axdx=\frac{1}{3}x^3 \csc^{-1} ax+\frac{x^2}{6a}\sqrt{1-\frac{1}{a^2x^2}}\\
&         +\frac{1}{6a^3} \tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\\
&(4) \displaystyle\int x^{m+1}\csc^{-1} axdx=\frac{x^{m+1} \csc^{-1} ax}{m+1}\\
&         +\frac{1}{a(m+1)}\displaystyle\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}dx (m≠-1)
\end{alignat}




<証明>
(1)から(4)まで、部分積分、分子から1を引いて加え積分を切り離す、及び下記の \((A)\) から \((C)\) の式を用いて解いています。
\begin{alignat}{2}
&(A) \{(a^2x^2-1)^{\frac{1}{2}}\}’=\frac{1}{2}(a^2x^2-1)^{-\frac{1}{2}}\cdot 2a^2x=\frac{a^2x}{\sqrt{a^2x^2-1}}\\
&\\
&(B) \left( \tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\right)’\\
&=\frac{1}{1-\left(1-\frac{1}{a^2x^2}\right)}\cdot \frac{1}{a} \cdot \frac{\frac{1}{2}\cdot \frac{1}{\sqrt{a^2x^2-1}} \cdot 2a^2x \cdot x -\sqrt{a^2x^2-1}}{x^2}\\
&=a^2x^2 \cdot \frac{1}{a}\cdot \frac{a^2x^2-(a^2x^2-1)}{x^2\sqrt{a^2x^2-1}}=\frac{a}{\sqrt{a^2x^2-1}}\\
&\\
&        \displaystyle\int \frac{a}{\sqrt{a^2x^2-1}}dx=\tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\\
&\\
&(C) \displaystyle\int \sqrt{a^2x^2-1}dx=x\sqrt{a^2x^2-1}-\displaystyle\int x \cdot \frac{1}{2}\cdot \frac{2a^2x}{\sqrt{a^2x^2-1}}dx\\
&                  =x\sqrt{a^2x^2-1}-\displaystyle\int \frac{a^2x^2}{\sqrt{a^2x^2-1}}dx\\
&                  =x\sqrt{a^2x^2-1}-\displaystyle\int \frac{a^2x^2-1+1}{\sqrt{a^2x^2-1}}dx\\
&                  =x\sqrt{a^2x^2-1}-\displaystyle\int \sqrt{a^2x^2-1}dx-\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&    2\displaystyle\int \sqrt{a^2x^2-1}dx=x\sqrt{a^2x^2-1}-\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&     \displaystyle\int \sqrt{a^2x^2-1}dx=\frac{1}{2}x\sqrt{a^2x^2-1}-\frac{1}{2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx
\end{alignat}





\begin{alignat}{2}
&(1) \displaystyle\int \csc^{-1} axdx=x \csc^{-1} ax+\displaystyle\int x \cdot \frac{1}{x\sqrt{a^2x^2-1}}dx\\
&              =x \csc^{-1} ax+\frac{1}{a}\displaystyle\int \frac{a}{\sqrt{a^2x^2-1}}dx\\
&              =x \csc^{-1} ax+\frac{1}{a} \tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\\
&\\
&\\
&\\
&\\
&(2) \displaystyle\int x \csc^{-1}axdx=\frac{1}{2}x^2 \sec^{-1}ax+\displaystyle\int \frac{1}{2}x^2\cdot \frac{1}{x\sqrt{a^2x^2-1}}dx\\
&                =\frac{1}{2}x^2 \csc^{-1}ax+\frac{1}{2a^2}\displaystyle\int \frac{a^2x}{\sqrt{a^2x^2-1}}dx\\
&                =\frac{1}{2}x^2 \csc^{-1}ax+\frac{1}{2a^2}\displaystyle\int \{(a^2x^2-1)^{\frac{1}{2}}\}’dx\\
&                =\frac{1}{2}x^2 \csc^{-1}ax+\frac{1}{2a^2}\sqrt{a^2x^2-1}dx\\
&                =\frac{1}{2}x^2 \csc^{-1}ax+\frac{ax}{2a^2}\sqrt{1-\frac{1}{a^2x^2}}dx\\
&                =\frac{1}{2}x^2 \csc^{-1} ax+\frac{x}{2a}\sqrt{1-\frac{1}{a^2x^2}}\\
&\\
&\\
&\\
&\\
&(3) \displaystyle\int x^2 \csc^{-1} ax\\
&=\frac{1}{3}x^3 \sec^{-1} ax+\displaystyle\int \frac{1}{3}x^3 \cdot \frac{1}{x\sqrt{a^2x^2-1}}\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{1}{3a^2}\displaystyle\int \frac{a^2x^2}{\sqrt{a^2x^2-1}}\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{1}{3a^2}\displaystyle\int \frac{a^2x^2-1+1}{\sqrt{a^2x^2-1}}\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{1}{3a^2}\displaystyle\int \sqrt{a^2x^2-1}dx+\frac{1}{3a^2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{1}{3a^2}\left(\frac{1}{2}x\sqrt{a^2x^2-1}-\frac{1}{2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\right)+\frac{1}{3a^2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{x}{6a^2}\sqrt{a^2x^2-1}-\frac{1}{6a^2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx+\frac{1}{3a^2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{x}{6a^2}\cdot ax \sqrt{1-\frac{1}{a^2x^2}}+\frac{1}{6a^2}\displaystyle\int \frac{1}{\sqrt{a^2x^2-1}}dx\\
&=\frac{1}{3}x^3 \csc^{-1} ax+\frac{x^2}{6a}\sqrt{1-\frac{1}{a^2x^2}}+\frac{1}{6a^3} \tanh^{-1} \sqrt{1-\frac{1}{a^2x^2}}\\
&\\
&\\
&\\
&\\
&(4) \displaystyle\int x^m \csc^{-1} axdx=\frac{1}{m+1}x^{m+1} \csc^{-1} ax+\displaystyle\int \frac{1}{m+1}x^{m+1}\cdot \frac{1}{x\sqrt{a^2x^2-1}}dx\\
&                 =\frac{x^{m+1} \csc^{-1} ax}{m+1}+\frac{1}{m+1}\displaystyle\int \frac{x^m}{ax\sqrt{1-\frac{1}{a^2x^2}}}dx\\
&                 =\frac{x^{m+1} \csc^{-1} ax}{m+1}+\frac{1}{a(m+1)}\displaystyle\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}dx
\end{alignat}

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