三角関数と双曲線関数の積分
三角関数の積分
(1)
\[ \int f(\sin x)\cos xdx=\int f(t)dt\cmt{t=\sin x} \](2)
\[ \int f(\cos x)\sin xdx=-\int f(t)dt\cmt{t=\cos x} \](3)
\[ \int f(\tan x)\frac{1}{\cos^{2}x}dx=\int f(t)dt\cmt{t=\tan x} \](4)ワイエルシュトラス置換
\[ \int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cmt{t=\tan\frac{x}{2}} \](1)
\begin{align*} \int f(\sin x)\cos xdx & =\int f(\sin x)d\left(\sin x\right)\\ & =\int f(t)dt\cmt{t=\sin x} \end{align*}(2)
\begin{align*} \int f(\cos x)\sin xdx & =-\int f(\cos x)d\left(\cos x\right)\\ & =-\int f(t)dt\cmt{t=\cos x} \end{align*}(3)
\begin{align*} \int f(\tan x)\frac{1}{\cos^{2}x}dx & =\int f(\tan x)d\left(\tan x\right)\\ & =\int f(t)dt\cmt{t=\tan x} \end{align*}(4)
\begin{align*} \int f(\cos x,\sin x)dx & =\int f\left(\cos^{2}\left(\frac{x}{2}\right)-\sin^{2}\left(\frac{x}{2}\right),2\sin\frac{x}{2}\cos\frac{x}{2}\right)dx\\ & =\int f\left(\frac{1-\tan^{2}\frac{x}{2}}{1+\tan^{2}\frac{x}{2}},\frac{2\tan\frac{x}{2}}{1+\tan^{2}\frac{x}{2}}\right)dx\\ & =\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)d\left(2\tan^{\bullet}t\right)\cmt{t=\tan\frac{x}{2}}\\ & =\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt \end{align*}双曲線関数の積分
(1)
\[ \int f(\sinh x)\cosh xdx=\int f(t)dt\cmt{t=\sinh x} \](2)
\[ \int f(\cosh x)\sinh xdx=\int f(t)dt\cmt{t=\cosh x} \](3)
\[ \int f(\tanh x)\frac{1}{\cosh^{2}x}dx=\int f(t)dt\cmt{t=\tanh x} \](4)
\[ \int f(\cosh x,\sinh x)dx=\int f\left(\frac{1+t^{2}}{1-t^{2}},\frac{2t}{1-t^{2}}\right)\frac{2}{1-t^{2}}dt\cmt{t=\tanh\frac{x}{2}} \](1)
\begin{align*} \int f(\sinh x)\cosh xdx & =\int f(\sinh x)d\left(\sinh x\right)\\ & =\int f(t)dt\cmt{t=\sinh x} \end{align*}(2)
\begin{align*} \int f(\cosh)\sinh xdx & =\int f(\cosh x)d\left(\cosh x\right)\\ & =\int f(t)dt\cmt{t=\cosh x} \end{align*}(3)
\begin{align*} \int f(\tanh x)\frac{1}{\cosh^{2}x}dx & =\int f(\tanh x)d\left(\tanh x\right)\\ & =\int f(t)dt\cmt{t=\tanh x} \end{align*}(4)
\begin{align*} \int f(\cosh x,\sinh x)dx & =\int f\left(\cosh^{2}\left(\frac{x}{2}\right)+\sinh^{2}\left(\frac{x}{2}\right),2\sinh\frac{x}{2}\cosh\frac{x}{2}\right)dx\\ & =\int f\left(\frac{1+\tanh^{2}\frac{x}{2}}{1-\tanh^{2}\frac{x}{2}},\frac{2\tanh\frac{x}{2}}{1-\tanh^{2}\frac{x}{2}}\right)dx\\ & =\int f\left(\frac{1+t^{2}}{1-t^{2}},\frac{2t}{1-t^{2}}\right)d\left(2\tanh^{\bullet}t\right)\cmt{t=\tanh\frac{x}{2}}\\ & =\int f\left(\frac{1+t^{2}}{1-t^{2}},\frac{2t}{1-t^{2}}\right)\frac{2}{1-t^{2}}dt \end{align*}
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正弦と余弦のべき乗の積の積分の超幾何関数表示
\[
\int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx=\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1-\beta}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C
\]
3角関数のべき乗の積分の超幾何関数表示
\[
\int\sin^{\alpha}\left(x\right)dx=\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C
\]
逆3角関数と逆双曲線関数の主値と2乗のルート
\[
\sin^{\bullet}\sin z=z\Rightarrow\sqrt{\cos^{2}z}=\cos z
\]
3角関数と逆3角関数・双曲線関数と逆双曲線関数の関係
\[
\sin^{\bullet}\sin z=z\Leftrightarrow\cos^{\bullet}\cos\left(\frac{\pi}{2}-z\right)=\frac{\pi}{2}-z
\]