(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。

(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。