逆三角関数と逆双曲線関数の積分

逆三角関数の積分

(1)

\[ \int\sin^{\circ}xdx=x\sin^{\circ}x+\sqrt{1-x^{2}} \]

(2)

\[ \int\cos^{\circ}xdx=x\cos^{\circ}x-\sqrt{1-x^{2}} \]

(3)

\[ \int\tan^{\circ}xdx=x\tan^{\circ}x-\frac{\log(x^{2}+1)}{2} \]

(4)

\[ \int\sin^{-1,\circ}xdx=x\sin^{-1,\circ}x+\log\left|x+x\sqrt{1-x^{-2}}\right| \]
\[ \int\sin^{-1,\circ}xdx=x\cos^{-1,\circ}x+\tanh^{\circ}\sqrt{1-x^{-2}} \]

(5)

\[ \int\cos^{-1,\circ}xdx=x\cos^{-1,\circ}x-\log\left|x+x\sqrt{1-x^{-2}}\right| \]

\[ \int\cos^{-1,\circ}xdx=x\cos^{-1,\circ}x-\tanh^{\circ}\sqrt{1-x^{-2}} \]

(6)

\[ \int\tan^{-1,\circ}xdx=x\tan^{-1,\circ}x+\frac{\log(x^{2}+1)}{2} \]

(1)

\begin{align*} \int\sin^{\circ}xdx & =x\sin^{\circ}x-\int\frac{x}{\sqrt{1-x^{2}}}dx\\ & =x\sin^{\circ}x+\sqrt{1-x^{2}} \end{align*}

(2)

\begin{align*} \int\cos^{\circ}xdx & =x\cos^{\circ}x+\int\frac{x}{\sqrt{1-x^{2}}}dx\\ & =x\cos^{\circ}x-\sqrt{1-x^{2}} \end{align*}

(3)

\begin{align*} \int\tan^{\circ}xdx & =x\tan^{\circ}x-\int\frac{x}{1+x^{2}}dx\\ & =x\tan^{\circ}x-\frac{\log(x^{2}+1)}{2} \end{align*}

(4)

\begin{align*} \int\sin^{-1,\circ}xdx & =x\sin^{-1,\circ}x+\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\circ}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,\circ}x+\int\frac{1}{x+x\sqrt{1-x^{-2}}}\left(1+\frac{1}{\sqrt{1-x^{-2}}}\right)dx\\ & =x\sin^{-1,\circ}x+\int\frac{\left(x+x\sqrt{1-x^{-2}}\right)'}{x+x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\circ}x+\log\left|x+x\sqrt{1-x^{-2}}\right| \end{align*}

(4)-2

\begin{align*} \int\sin^{-1,\circ}xdx & =x\sin^{-1,\circ}x+\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\sin^{-1,\circ}x+\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1-x^{-2}}\\ & =x\sin^{-1,\circ}x+\tanh^{\circ}y\\ & =x\sin^{-1,\circ}x+\tanh^{\circ}\sqrt{1-x^{-2}} \end{align*}

(5)

\begin{align*} \int\cos^{-1,\circ}xdx & =x\cos^{-1,\circ}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\circ}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\circ}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,\circ}x-\int\frac{1}{x+x\sqrt{1-x^{-2}}}\left(1+\frac{1}{\sqrt{1-x^{-2}}}\right)dx\\ & =x\cos^{-1,\circ}x-\int\frac{\left(x+x\sqrt{1-x^{-2}}\right)'}{x+x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\circ}x-\log\left|x+x\sqrt{1-x^{-2}}\right| \end{align*}

(5)-2

\begin{align*} \int\cos^{-1,\circ}xdx & =x\cos^{-1,\circ}x-\int\frac{1}{x\sqrt{1-x^{-2}}}dx\\ & =x\cos^{-1,\circ}x-\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1-x^{-2}}\\ & =x\cos^{-1,\circ}x-\tanh^{\circ}y\\ & =x\cos^{-1,\circ}x-\tanh^{\circ}\sqrt{1-x^{-2}} \end{align*}

(6)

\begin{align*} \int\tan^{-1,\circ}xdx & =x\tan^{-1,\circ}x+\int\frac{x}{1+x^{2}}dx\\ & =x\tan^{-1,\circ}x+\frac{\log(x^{2}+1)}{2} \end{align*}

逆走曲線関数の積分

(1)

\[ \int\sinh^{\circ}xdx=x\sinh^{\circ}x-\sqrt{1+x^{2}} \]

(2)

\[ \int\cosh^{\circ}xdx=x\cosh^{\circ}x-\sqrt{x^{2}-1} \]

(3)

\[ \int\tanh^{\circ}xdx=x\tanh^{\circ}x+\frac{\log(1-x^{2})}{2} \]

(4)

\[ \int\sinh^{-1,\circ}xdx=x\sinh^{-1,\circ}x+\log\left|x+x\sqrt{1+x^{-2}}\right| \]
\[ \int\sinh^{-1,\circ}xdx=x\sinh^{-1,\circ}x+\tanh^{\circ}\sqrt{1+x^{-2}} \]

(5)

\[ \int\cosh^{-1,\circ}xdx=x\cosh^{-1,\circ}x-\tan^{\circ}\sqrt{x^{-2}-1} \]

\[ \int\cosh^{-1,\circ}xdx=x\cosh^{-1,\circ}x+\sin^{\circ}x \]

(6)

\[ \int\tanh^{-1,\circ}xdx=x\tanh^{-1,\circ}x+\frac{\log(x^{2}-1)}{2} \]

(1)

\begin{align*} \int\sinh^{\circ}xdx & =x\sinh^{\circ}x-\int\frac{x}{\sqrt{1+x^{2}}}dx\\ & =x\sinh^{\circ}x-\sqrt{1+x^{2}} \end{align*}

(2)

\begin{align*} \int\cosh^{\circ}xdx & =x\cosh^{\circ}x-\int\frac{x}{\sqrt{x^{2}-1}}dx\\ & =x\cosh^{\circ}x-\sqrt{x^{2}-1} \end{align*}

(3)

\begin{align*} \int\tanh^{\circ}xdx & =x\tanh^{\circ}x-\int\frac{x}{1-x^{2}}dx\\ & =x\tanh^{\circ}x+\frac{\log(1-x^{2})}{2} \end{align*}

(4)

\begin{align*} \int\sinh^{-1,\circ}xdx & =x\sinh^{-1,\circ}x+\int\frac{1}{x\sqrt{1+x^{-2}}}dx\\ & =x\sinh^{-1,\circ}x+\log\left|x+x\sqrt{1+x^{-2}}\right| \end{align*}

(4)-2

\begin{align*} \int\sinh^{-1,\circ}xdx & =x\sinh^{-1,\circ}x+\int\frac{1}{x\sqrt{1+x^{-2}}}dx\\ & =x\sinh^{-1,\circ}x+\int\frac{1}{1-y^{2}}dy\qquad,\qquad y=\sqrt{1+x^{-2}}\\ & =x\sinh^{-1,\circ}x+\tanh^{\circ}y\\ & =x\sinh^{-1,\circ}x+\tanh^{\circ}\sqrt{1+x^{-2}} \end{align*}

(5)

\begin{align*} \int\cosh^{-1,\circ}xdx & =x\cosh^{-1,\circ}x+\int\frac{1}{x\sqrt{x^{-2}-1}}dx\\ & =x\cosh^{-1,\circ}x-\int\frac{1}{1+y^{2}}dy\qquad,\qquad y=\sqrt{x^{-2}-1}\\ & =x\cosh^{-1,\circ}x-\tan^{\circ}y\\ & =x\cosh^{-1,\circ}x-\tan^{\circ}\sqrt{x^{-2}-1} \end{align*}

(5)-2

\begin{align*} \int\cosh^{-1,\circ}xdx & =x\cosh^{-1,\circ}x+\int\frac{1}{x\sqrt{x^{-2}-1}}dx\\ & =x\cosh^{-1,\circ}x+\int\frac{1}{\sqrt{1-x^{2}}}dx\qquad,\qquad\def(\cosh^{-1,\circ})=(0,1]\\ & =x\cosh^{-1,\circ}x+\sin^{\circ}x \end{align*}

(6)

\begin{align*} \int\tanh^{-1,\circ}xdx & =x\tanh^{-1,\circ}x-\int\frac{x}{1-x^{2}}dx\\ & =x\tanh^{-1,\circ}x+\frac{\log(x^{2}-1)}{2} \end{align*}

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逆三角関数と逆双曲線関数の積分

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