三角関数と双曲線関数の積分

三角関数の積分

(1)

\[ \int f(\sin x)\cos xdx=\int f(t)dt\cnd{t=\sin x} \]

(2)

\[ \int f(\cos x)\sin xdx=-\int f(t)dt\cnd{t=\cos x} \]

(3)

\[ \int f(\tan x)\frac{1}{\cos^{2}x}dx=\int f(t)dt\cnd{t=\tan x} \]

(4)

\[ \int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cnd{t=\tan\frac{x}{2}} \]

(1)

\begin{align*} \int f(\sin x)\cos xdx & =\int f(\sin x)d\left(\sin x\right)\\ & =\int f(t)dt\cnd{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\cnd{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\cnd{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 x\cos x\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^{\circ}t\right)\cnd{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)

\[ \int f(\sinh x)\cosh xdx=\int f(t)dt\cnd{t=\sinh x} \]

(2)

\[ \int f(\cosh x)\sinh xdx=\int f(t)dt\cnd{t=\cosh x} \]

(3)

\[ \int f(\tanh x)\frac{1}{\cosh^{2}x}dx=\int f(t)dt\cnd{t=\tanh x} \]

(4)

\[ \int f(\cosh x,\sinh x)dx==\int f\left(\frac{1+t^{2}}{1-t^{2}},\frac{2t}{1-t^{2}}\right)\frac{2}{1-t^{2}}dt\cnd{t=\tanh\frac{x}{2}} \]

(1)

\begin{align*} \int f(\sinh x)\cosh xdx & =\int f(\sinh x)d\left(\sinh x\right)\\ & =\int f(t)dt\cnd{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\cnd{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\cnd{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 x\cosh x\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^{\circ}t\right)\cnd{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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三角関数と双曲線関数の積分

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