逆三角関数と逆双曲線関数の積分
逆三角関数の積分
(1)
\[ \int\sin^{\bullet}xdx=x\sin^{\bullet}x+\sqrt{1-x^{2}} \]
(2)
\[ \int\cos^{\bullet}xdx=x\cos^{\bullet}x-\sqrt{1-x^{2}} \]
(3)
\[ \int\tan^{\bullet}xdx=x\tan^{\bullet}x-\frac{\log(x^{2}+1)}{2} \]
(4)
\[
\int\sin^{-1,\bullet}xdx=x\sin^{-1,\bullet}x+\log\left|x+x\sqrt{1-x^{-2}}\right|
\]
\[
\int\sin^{-1,\bullet}xdx=x\cos^{-1,\bullet}x+\tanh^{\bullet}\sqrt{1-x^{-2}}
\]
(5)
\[ \int\cos^{-1,\bullet}xdx=x\cos^{-1,\bullet}x-\log\left|x+x\sqrt{1-x^{-2}}\right| \]
\[ \int\cos^{-1,\bullet}xdx=x\cos^{-1,\bullet}x-\tanh^{\bullet}\sqrt{1-x^{-2}} \]
(6)
\[ \int\tan^{-1,\bullet}xdx=x\tan^{-1,\bullet}x+\frac{\log(x^{2}+1)}{2} \]
(1)
\begin{align*} \int\sin^{\bullet}xdx & =x\sin^{\bullet}x-\int\frac{x}{\sqrt{1-x^{2}}}dx\\ & =x\sin^{\bullet}x+\sqrt{1-x^{2}} \end{align*}
(2)
\begin{align*} \int\cos^{\bullet}xdx & =x\cos^{\bullet}x+\int\frac{x}{\sqrt{1-x^{2}}}dx\\ & =x\cos^{\bullet}x-\sqrt{1-x^{2}} \end{align*}
(3)
\begin{align*} \int\tan^{\bullet}xdx & =x\tan^{\bullet}x-\int\frac{x}{1+x^{2}}dx\\ & =x\tan^{\bullet}x-\frac{\log(x^{2}+1)}{2} \end{align*}
(4)
\begin{align*} \int\sin^{-1,\bullet}xdx & =x\sin^{-1,\bullet}x+\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\bullet}x+\int\frac{1}{x+x\sqrt{1-x^{-2}}}\frac{x+x\sqrt{1-x^{-2}}}{x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\bullet}x+\int\frac{1}{x+x\sqrt{1-x^{-2}}}\left(1+\frac{1}{\sqrt{1-x^{-2}}}\right)dx\\ & =x\sin^{-1,\bullet}x+\int\frac{\left(x+x\sqrt{1-x^{-2}}\right)'}{x+x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\bullet}x+\log\left|x+x\sqrt{1-x^{-2}}\right| \end{align*}
(4)-2
\begin{align*} \int\sin^{-1,\bullet}xdx & =x\sin^{-1,\bullet}x+\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\bullet}x+\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1-x^{-2}}\\ & =x\sin^{-1,\bullet}x+\tanh^{\bullet}y\\ & =x\sin^{-1,\bullet}x+\tanh^{\bullet}\sqrt{1-x^{-2}} \end{align*}
(5)
\begin{align*} \int\cos^{-1,\bullet}xdx & =x\cos^{-1,\bullet}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\bullet}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\bullet}x-\int\frac{1}{x+x\sqrt{1-x^{-2}}}\frac{x+x\sqrt{1-x^{-2}}}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\bullet}x-\int\frac{1}{x+x\sqrt{1-x^{-2}}}\left(1+\frac{1}{\sqrt{1-x^{-2}}}\right)dx\\ & =x\cos^{-1,\bullet}x-\int\frac{\left(x+x\sqrt{1-x^{-2}}\right)'}{x+x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\bullet}x-\log\left|x+x\sqrt{1-x^{-2}}\right| \end{align*}
(5)-2
\begin{align*} \int\cos^{-1,\bullet}xdx & =x\cos^{-1,\bullet}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\bullet}x-\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1-x^{-2}}\\ & =x\cos^{-1,\bullet}x-\tanh^{\bullet}y\\ & =x\cos^{-1,\bullet}x-\tanh^{\bullet}\sqrt{1-x^{-2}} \end{align*}
(6)
\begin{align*} \int\tan^{-1,\bullet}xdx & =x\tan^{-1,\bullet}x+\int\frac{x}{1+x^{2}}dx\\ & =x\tan^{-1,\bullet}x+\frac{\log(x^{2}+1)}{2} \end{align*}
逆走曲線関数の積分
(1)
\[ \int\sinh^{\bullet}xdx=x\sinh^{\bullet}x-\sqrt{1+x^{2}} \]
(2)
\[ \int\cosh^{\bullet}xdx=x\cosh^{\bullet}x-\sqrt{x^{2}-1} \]
(3)
\[ \int\tanh^{\bullet}xdx=x\tanh^{\bullet}x+\frac{\log(1-x^{2})}{2} \]
(4)
\[
\int\sinh^{-1,\bullet}xdx=x\sinh^{-1,\bullet}x+\log\left|x+x\sqrt{1+x^{-2}}\right|
\]
\[
\int\sinh^{-1,\bullet}xdx=x\sinh^{-1,\bullet}x+\tanh^{\bullet}\sqrt{1+x^{-2}}
\]
(5)
\[ \int\cosh^{-1,\bullet}xdx=x\cosh^{-1,\bullet}x-\tan^{\bullet}\sqrt{x^{-2}-1} \]
\[ \int\cosh^{-1,\bullet}xdx=x\cosh^{-1,\bullet}x+\sin^{\bullet}x \]
(6)
\[ \int\tanh^{-1,\bullet}xdx=x\tanh^{-1,\bullet}x+\frac{\log(x^{2}-1)}{2} \]
(1)
\begin{align*} \int\sinh^{\bullet}xdx & =x\sinh^{\bullet}x-\int\frac{x}{\sqrt{1+x^{2}}}dx\\ & =x\sinh^{\bullet}x-\sqrt{1+x^{2}} \end{align*}
(2)
\begin{align*} \int\cosh^{\bullet}xdx & =x\cosh^{\bullet}x-\int\frac{x}{\sqrt{x^{2}-1}}dx\\ & =x\cosh^{\bullet}x-\sqrt{x^{2}-1} \end{align*}
(3)
\begin{align*} \int\tanh^{\bullet}xdx & =x\tanh^{\bullet}x-\int\frac{x}{1-x^{2}}dx\\ & =x\tanh^{\bullet}x+\frac{\log(1-x^{2})}{2} \end{align*}
(4)
\begin{align*} \int\sinh^{-1,\bullet}xdx & =x\sinh^{-1,\bullet}x+\int\frac{1}{x\sqrt{1+x^{-2}}}dx\\ & =x\sinh^{-1,\bullet}x+\log\left|x+x\sqrt{1+x^{-2}}\right| \end{align*}
(4)-2
\begin{align*} \int\sinh^{-1,\bullet}xdx & =x\sinh^{-1,\bullet}x+\int\frac{1}{x\sqrt{1+x^{-2}}}dx\\ & =x\sinh^{-1,\bullet}x+\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1+x^{-2}}\\ & =x\sinh^{-1,\bullet}x+\tanh^{\bullet}y\\ & =x\sinh^{-1,\bullet}x+\tanh^{\bullet}\sqrt{1+x^{-2}} \end{align*}
(5)
\begin{align*} \int\cosh^{-1,\bullet}xdx & =x\cosh^{-1,\bullet}x+\int\frac{1}{x\sqrt{x^{-2}-1}}dx\\ & =x\cosh^{-1,\bullet}x-\int\frac{1}{1+y^{2}}dy\qquad,\qquad y=\sqrt{x^{-2}-1}\\ & =x\cosh^{-1,\bullet}x-\tan^{\bullet}y\\ & =x\cosh^{-1,\bullet}x-\tan^{\bullet}\sqrt{x^{-2}-1} \end{align*}
(5)-2
\begin{align*} \int\cosh^{-1,\bullet}xdx & =x\cosh^{-1,\bullet}x+\int\frac{1}{x\sqrt{x^{-2}-1}}dx\\ & =x\cosh^{-1,\bullet}x+\int\frac{1}{\sqrt{1-x^{2}}}dx\qquad,\qquad\def(\cosh^{-1,\bullet})=(0,1]\\ & =x\cosh^{-1,\bullet}x+\sin^{\bullet}x \end{align*}
(6)
\begin{align*} \int\tanh^{-1,\bullet}xdx & =x\tanh^{-1,\bullet}x-\int\frac{x}{1-x^{2}}dx\\ & =x\tanh^{-1,\bullet}x+\frac{\log(x^{2}-1)}{2} \end{align*}
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