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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逆三角関数と逆双曲線関数の積分表示
\[
\sin^{\bullet}x=\int_{0}^{x}\frac{1}{\sqrt{1-t^{2}}}dt
\]
三角関数を正接の半角、双曲線関数を双曲線正接の半角で表す。
\[
\sin z=\frac{2\tan\frac{z}{2}}{1+\tan^{2}\frac{z}{2}}
\]
三角関数と双曲線関数の積分
\[
\int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cmt{t=\tan\frac{x}{2}}
\]
三角関数と双曲線関数の実部と虚部
\[
\sin z=\sin\left(\Re\left(z\right)\right)\cosh\left(\Im\left(z\right)\right)+i\cos\left(\Re\left(z\right)\right)\sinh\left(\Im\left(z\right)\right)
\]