ルートの中に2乗を含む積分
ルートの中に2乗を含む積分
(1)
\[ \int f\left(\sqrt{a^{2}-x^{2}}\right)dx=a\int f\left(a\cos t\right)\cos tdt\cnd{x=a\sin t} \]
(1)-2
\[ \int f\left(\sqrt{a^{2}-x^{2}}\right)dx=a\int f\left(\frac{a}{\cosh t}\right)\frac{1}{\cosh^{2}t}dt\cnd{x=a\tanh t} \]
(2)
\[ \int f\left(\sqrt{a^{2}+x^{2}}\right)dx=a\int f\left(a\cosh t\right)\cosh tdt\cnd{x=a\sinh t} \]
(2)-2
\[ \int f\left(\sqrt{a^{2}+x^{2}}\right)dx=a\int f\left(\frac{a}{\cos t}\right)\frac{1}{\cos^{2}t}dt\cnd{x=a\tan t} \]
(2)-3
\[ \int f\left(\sqrt{a^{2}+x^{2}}\right)dx=-\frac{1}{2}\int f\left(\frac{t^{2}+a^{2}}{2t}\right)\frac{t^{2}+a^{2}}{t^{2}}dt\cnd{t=x+\sqrt{x^{2}+a^{2}}} \]
(3)
\[ \int f\left(\sqrt{x^{2}-a^{2}}\right)dx=a\int f\left(a\sinh t\right)\sinh tdt\cnd{x=a\cosh t} \]
(3)-2
\[ \int f\left(\sqrt{x^{2}-a^{2}}\right)dx=-a\int f\left(\frac{a}{\sinh t}\right)\frac{1}{\sinh^{2}t}dt\cnd{x=a\tanh^{-1}t} \]
(3)-3
\[ \int f\left(\sqrt{x^{2}-a^{2}}\right)dx=\frac{1}{2}\int f\left(\frac{t^{2}-a^{2}}{2t}\right)\frac{t^{2}-a^{2}}{t}dt\cnd{t=x+\sqrt{x^{2}-a^{2}}} \]
(1)
\begin{align*} \int f\left(\sqrt{a^{2}-x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}-a^{2}\sin^{2}t}\right)d\left(a\sin t\right)\cnd{x=a\sin t}\\ & =a\int f\left(a\cos t\right)\cos tdt \end{align*}
(1)-2
\begin{align*} \int f\left(\sqrt{a^{2}-x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}-a^{2}\tanh^{2}t}\right)d\left(a\tanh t\right)\cnd{x=a\tanh t}\\ & =a\int f\left(\frac{a}{\cosh t}\right)\frac{1}{\cosh^{2}t}dt \end{align*}
(2)
\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}+a^{2}\sinh^{2}t}\right)d\left(a\sinh t\right)\cnd{x=a\sinh t}\\ & =a\int f\left(a\cosh t\right)\cosh tdt \end{align*}
(2)-2
\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}+a^{2}\tan^{2}t}\right)d\left(a\tan t\right)\cnd{x=a\tan t}\\ & =a\int f\left(\frac{a}{\cos t}\right)\frac{1}{\cos^{2}t}dt \end{align*}
(2)-3
\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\left|\frac{t^{2}+a^{2}}{2t}\right|\right)d\left(\frac{a^{2}-t^{2}}{2t}\right)\cnd{t=x+\sqrt{x^{2}+a^{2}}}\\ & =-\frac{1}{2}\int f\left(\frac{t^{2}+a^{2}}{2t}\right)\frac{t^{2}+a^{2}}{t^{2}}dt \end{align*}
(3)
\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\sqrt{a^{2}\cosh^{2}t-a^{2}}\right)d\left(a\cosh t\right)\cnd{x=a\cosh t}\\ & =a\int f\left(a\sinh t\right)\sinh tdt \end{align*}
(3)-2
\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\sqrt{a^{2}\tanh^{-2}-a^{2}}\right)d\left(a\tanh^{-1}t\right)\cnd{x=a\tanh^{-1}t}\\ & =-a\int f\left(\frac{a}{\sinh t}\right)\frac{1}{\sinh^{2}t}dt \end{align*}
(3)-3
\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\left|\frac{t^{2}-a^{2}}{2t}\right|\right)d\left(\frac{t^{2}+a^{2}}{2t}\right)\cnd{t=x+\sqrt{x^{2}-a^{2}}}\\ & =\frac{1}{2}\int f\left(\frac{t^{2}-a^{2}}{2t}\right)\frac{t^{2}-a^{2}}{t}dt \end{align*}
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