三角関数と双曲線関数の積分

三角関数の積分

(1)

\[ \int\sin xdx=-\cos x \]

(2)

\[ \int\cos xdx=\sin x \]

(3)

\[ \int\tan xdx=\log\left|\cos^{-1}x\right| \]

(4)

\[ \int\sin^{-1}xdx=-\log\left|\sin^{-1}x+\tan^{-1}x\right| \]

(5)

\[ \int\cos^{-1}xdx=\log\left|\tan x+\cos^{-1}x\right| \]

(6)

\[ \int\tan^{-1}xdx=\log\left|\sin x\right| \]

(1)

\begin{align*} \int\sin xdx & =-\int(\cos x)'dx\\ & =-\cos x \end{align*}

(2)

\begin{align*} \int\cos xdx & =\int(\sin x)'dx\\ & =\sin x \end{align*}

(3)

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

(4)

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

(5)

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

(6)

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

双曲線関数の積分

(1)

\[ \int\sinh xdx=\cosh x \]

(2)

\[ \int\cosh xdx=\sinh x \]

(3)

\[ \int\tanh xdx=\log\left|\cosh x\right| \]

(4)

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

(5)

\[ \int\cosh^{-1}xdx=\tan^{\circ}\sinh x \]

or
\[ \int\cosh^{-1}xdx=\sin^{\circ}\tanh x \]

or

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

(6)

\[ \int\tan^{-1}xdx=\log\left|\sinh x\right| \]

(1)

\begin{align*} \int\sinh xdx & =-\int\sin(ix)idx\\ & =\cos(ix)\\ & =\cosh x \end{align*}

(2)

\begin{align*} \int\cosh xdx & =-i\int\cos(ix)idx\\ & =-i\sin(ix)\\ & =\sinh x \end{align*}

(3)

\begin{align*} \int\tanh xdx & =-\int\tan(ix)idx\\ & =-\log\left|\cos^{-1}(ix)\right|\\ & =\log\left|\cosh x\right| \end{align*}

(4)

\begin{align*} \int\sinh^{-1}xdx & =\int\sin^{-1}(ix)idx\\ & ==-\log\left|\sin^{-1}(ix)+\tan^{-1}(ix)\right|\\ & =-\log\left|\sinh^{-1}x+\tanh^{-1}x\right| \end{align*}

(5)

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

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

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

(6)

\begin{align*} \int\tanh^{-1}xdx & =\int\tan^{-1}(ix)idx\\ & =\log\left|\sin(ix)\right|\\ & =\log\left|\sinh x\right| \end{align*}

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三角関数と双曲線関数の積分

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