三角関数と双曲線関数の冪乗積分漸化式
三角関数の冪乗積分漸化式
(1)
\[ \int\sin^{n}xdx=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}xdx\qquad(n\ne0) \](2)
\[ \int\sin^{n}xdx=\frac{1}{n+1}\cos x\sin^{n+1}x+\frac{n+2}{n+1}\int\sin^{n+2}xdx\qquad(n\ne-1) \](3)
\[ \int\cos^{n}xdx=\frac{1}{n}\sin x\cos^{n-1}x+\frac{n-1}{n}\int\cos^{n-2}xdx\qquad(n\ne0) \](4)
\[ \int\cos^{n}xdx=-\frac{1}{n+1}\sin x\cos^{n+1}x+\frac{n+2}{n+1}\int\cos^{n+2}xdx\qquad(n\ne-1) \](5)
\[ \int\tan^{n}xdx=\frac{1}{n-1}\tan^{n-1}x-\int\tan^{n-2}xdx\qquad(n\ne1) \](6)
\[ \int\tan^{n}xdx=\frac{1}{n+1}\tan^{n+1}x-\int\tan^{n+2}xdx\qquad(n\ne-1) \](1)
\begin{align*} \int\sin^{n}xdx & =\int\sin x\sin^{n-1}xdx\\ & =-\cos x\sin^{n-1}x+(n-1)\int\cos^{2}x\sin^{n-2}xdx\\ & =-\cos x\sin^{n-1}x+(n-1)\int(1-\sin^{2}x)\sin^{n-2}xdx\\ & =-\cos x\sin^{n-1}x+(n-1)\int\sin^{n-2}xdx-(n-1)\int\sin^{n}xdx\\ & =-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}xdx \end{align*}(2)
\begin{align*} \int\sin^{n}xdx & =\int(\cos^{2}x+\sin^{2}x)\sin^{n}xdx\\ & =\int\cos x\sin^{n}x(\sin x)'dx+\int\sin^{n+2}xdx\\ & =\frac{1}{n+1}\cos x\sin^{n+1}x+\frac{1}{n+1}\int\sin^{n+2}xdx+\int\sin^{n+2}xdx\\ & =\frac{1}{n+1}\cos x\sin^{n+1}x+\frac{n+2}{n+1}\int\sin^{n+2}xdx \end{align*}(2)-2
(1)で\(n\rightarrow n+2\)とすればいい。(3)
\begin{align*} \int\cos^{n}xdx & =\int\cos x\cos^{n-1}xdx\\ & =\sin x\cos^{n-1}x+(n-1)\int\sin^{2}x\cos^{n-2}xdx\\ & =\sin x\cos^{n-1}x+(n-1)\int(1-\cos^{2}x)\cos^{n-2}xdx\\ & =\sin x\cos^{n-1}x+(n-1)\int\cos^{n-2}xdx-(n-1)\int\cos^{n}xdx\\ & =\frac{1}{n}\sin x\cos^{n-1}x+\frac{n-1}{n}\int\cos^{n-2}xdx \end{align*}(3)-2
(1)で\(x=y+\frac{\pi}{2}\)とおくと、\[ \int\cos^{n}xdx=\frac{1}{n}\sin x\cos^{n-1}x+\frac{n-1}{n}\int\cos^{n-2}xdx \]
(4)
\begin{align*} \int\cos^{n}xdx & =\int(\cos^{2}x+\sin^{2}x)\cos^{n}xdx\\ & =\int\cos^{n+2}xdx-\int\sin x\cos^{n}x(\cos x)'dx\\ & =\int\cos^{n+2}xdx-\frac{1}{n+1}\sin x\cos^{n+1}x+\frac{1}{n+1}\int\cos^{n+2}xdx\\ & =-\frac{1}{n+1}\sin x\cos^{n+1}x+\frac{n+2}{n+1}\int\cos^{n+2}xdx \end{align*}(4)-2
(3)で\(n\rightarrow n+2\)とすればいい。(5)
\begin{align*} \int\tan^{n}xdx & =\int\tan^{n-2}x\tan^{2}xdx\\ & =\int\tan^{n-2}x\left(\cos^{-2}x-1\right)dx\\ & =\int\tan^{n-2}x(\tan x)'dx-\int\tan^{n-2}xdx\\ & =\frac{1}{n-1}\tan^{n-1}x-\int\tan^{n-2}xdx \end{align*}(6)
\begin{align*} \int\tan^{n}xdx & =\int\tan^{n}x(\cos^{-2}x-\tan^{2}x)dx\\ & =\int\tan^{n}x(\tan x)'dx-\int\tan^{n+2}xdx\\ & =\frac{1}{n+1}\tan^{n+1}x-\int\tan^{n+2}xdx \end{align*}(6)-2
(5)で\(n\rightarrow n+2\)とすればいい。双曲線関数の冪乗積分漸化式
(1)
\[ \int\sinh^{n}xdx=\frac{1}{n}\cosh x\sinh^{n-1}x-\frac{n-1}{n}\int\sinh^{n-2}xdx\qquad(n\ne0) \](2)
\[ \int\sinh^{n}xdx=\frac{1}{n+1}\cosh x\sinh^{n+1}x-\frac{n+2}{n+1}\int\sinh^{n+2}xdx\qquad(n\ne-1) \](3)
\[ \int\cosh^{n}xdx=\frac{1}{n}\sinh x\cosh^{n-1}x+\frac{n-1}{n}\int\cosh^{n-2}xdx\qquad(n\ne0) \](4)
\[ \int\cosh^{n}xdx=-\frac{1}{n+1}\sinh x\cosh^{n+1}x+\frac{n+2}{n+1}\int\cosh^{n+2}xdx\qquad(n\ne-1) \](5)
\[ \int\tanh^{n}xdx=-\frac{1}{n-1}\tanh^{n-1}x+\int\tanh^{n-2}xdx\qquad(n\ne1) \](6)
\[ \int\tanh^{n}xdx=\frac{1}{n+1}\tanh^{n+1}x+\int\tanh^{n+2}xdx\qquad(n\ne-1) \](1)
\begin{align*} \int\sinh^{n}xdx & =\int\sinh x\sinh^{n-1}xdx\\ & =\cosh x\sinh^{n-1}x-(n-1)\int\cosh^{2}x\sinh^{n-2}xdx\\ & =\cosh x\sinh^{n-1}x-(n-1)\int(1+\sinh^{2}x)\sinh^{n-2}xdx\\ & =\cosh x\sinh^{n-1}x-(n-1)\int\sinh^{n-2}xdx-(n-1)\int\sinh^{n}xdx\\ & =\frac{1}{n}\cosh x\sinh^{n-1}x-\frac{n-1}{n}\int\sinh^{n-2}xdx \end{align*}(2)
\begin{align*} \int\sinh^{n}xdx & =\int(\cosh^{2}x-\sinh^{2}x)\sinh^{n}xdx\\ & =\int\cosh x\sinh^{n}x(\sinh x)'dx-\int\sinh^{n+2}xdx\\ & =\frac{1}{n+1}\cosh x\sinh^{n+1}x-\frac{1}{n+1}\int\sinh^{n+2}xdx-\int\sinh^{n+2}xdx\\ & =\frac{1}{n+1}\cosh x\sinh^{n+1}x-\frac{n+2}{n+1}\int\sinh^{n+2}xdx \end{align*}(2)-2
(1)で\(n\rightarrow n+2\)とすればいい。(3)
\begin{align*} \int\cosh^{n}xdx & =\int\cosh x\cosh^{n-1}xdx\\ & =\sinh x\cosh^{n-1}x-(n-1)\int\sinh^{2}x\cosh^{n-2}xdx\\ & =\sinh x\cosh^{n-1}x-(n-1)\int(\cosh^{2}x-1)\cosh^{n-2}xdx\\ & =\sinh x\cosh^{n-1}x+(n-1)\int\cosh^{n-2}xdx-(n-1)\int\cosh^{n}xdx\\ & =\frac{1}{n}\sinh x\cosh^{n-1}x+\frac{n-1}{n}\int\cosh^{n-2}xdx \end{align*}(4)
\begin{align*} \int\cosh^{n}xdx & =\int(\cosh^{2}x-\sinh^{2}x)\cosh^{n}xdx\\ & =\int\cosh^{n+2}xdx-\int\sinh x\cosh^{n}x(\cosh x)'dx\\ & =\int\cosh^{n+2}xdx-\frac{1}{n+1}\sinh x\cosh^{n+1}x+\frac{1}{n+1}\int\cosh^{n+2}x\\ & =-\frac{1}{n+1}\sinh x\cosh^{n+1}x+\frac{n+2}{n+1}\int\cosh^{n+2}xdx \end{align*}(4)-2
(3)で\(n\rightarrow n+2\)とすればいい。(5)
\begin{align*} \int\tanh^{n}xdx & =\int\tanh^{n-2}x\tanh^{2}xdx\\ & =\int\tanh^{n-2}x\left(1-\cosh^{-2}x\right)dx\\ & =-\int\tanh^{n-2}x(\tanh x)'dx+\int\tanh^{n-2}xdx\\ & =-\frac{1}{n-1}\tanh^{n-1}x+\int\tanh^{n-2}xdx \end{align*}(6)
\begin{align*} \int\tanh^{n}xdx & =\int\tanh^{n}x(\cosh^{-2}x+\tanh^{2}x)dx\\ & =\int\tanh^{n}x(\tanh x)'dx+\int\tanh^{n+2}xdx\\ & =\frac{1}{n+1}\tanh^{n+1}x+\int\tanh^{n+2}xdx \end{align*}(6)-2
(5)で\(n\rightarrow n+2\)とすればいい。
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\[
\sin^{2}\frac{x}{2}=\frac{1-\cos x}{2}
\]
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\[
\sin A+\sin B+\sin C=4\sin\frac{B+C}{2}\sin\frac{C+A}{2}\sin\frac{A+B}{2}+\sin\left(A+B+C\right)
\]
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\[
\int\sin^{\bullet,n}xdx=x\sin^{\bullet,n}x+n\sqrt{1-x^{2}}\sin^{\bullet,n-1}x-n(n-1)\int\sin^{\bullet,n-2}xdx
\]
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\[
\tan\frac{z}{2}=\frac{\sin z}{1+\cos z}
\]