分母に2乗のルートがある積分

分母に2乗のルートがある積分

\begin{align*} \int\frac{1}{\sqrt{z^{2}+\alpha}}dz & =\frac{\sqrt{\alpha}\sqrt{\frac{z^{2}}{\alpha}+1}}{\sqrt{z^{2}+\alpha}}\sinh^{\circ}\frac{z}{\sqrt{\alpha}}+C\\ & =\tanh^{\circ}\frac{z}{\sqrt{z^{2}+\alpha}}+C\\ & =\frac{1}{2}\left(\Log\left(1+\frac{z}{\sqrt{z^{2}+\alpha}}\right)-\Log\left(1-\frac{z}{\sqrt{z^{2}+\alpha}}\right)\right)+C \end{align*}

(0-1)

\begin{align*} \int\frac{1}{\sqrt{z^{2}+\alpha}}dz & =\frac{\sqrt{\alpha}\sqrt{\frac{z^{2}}{\alpha}+1}}{\sqrt{z^{2}+\alpha}}\int\frac{1}{\sqrt{\alpha}\sqrt{\frac{z^{2}}{\alpha}+1}}dz\\ & =\frac{\sqrt{\alpha}\sqrt{\frac{z^{2}}{\alpha}+1}}{\sqrt{z^{2}+\alpha}}\int\frac{1}{\sqrt{\left(\frac{z}{\sqrt{\alpha}}\right)^{2}+1}}d\frac{z}{\sqrt{\alpha}}\\ & =\frac{\sqrt{\alpha}\sqrt{\frac{z^{2}}{\alpha}+1}}{\sqrt{z^{2}+\alpha}}\sinh^{\circ}\frac{z}{\sqrt{\alpha}}+C \end{align*}

(0-2)

\begin{align*} \int\frac{1}{\sqrt{z^{2}+\alpha}}dz & =\int\frac{t}{z}dz\cnd{t=\frac{z}{\sqrt{z^{2}+\alpha}}\;,\;z^{2}=\alpha\frac{t^{2}}{1-t^{2}}\;,\;dt=\alpha\left(\frac{t}{z}\right)^{3}dz}\\ & =\frac{1}{\alpha}\int\left(\frac{z}{t}\right)^{2}dt\\ & =\int\frac{1}{1-t^{2}}dt\\ & =\tanh^{\circ}t+C\\ & =\tanh^{\circ}\frac{z}{\sqrt{z^{2}+\alpha}}+C \end{align*}

(0-3)

\begin{align*} \int\frac{1}{\sqrt{z^{2}+\alpha}}dz & =\tanh^{\circ}\frac{z}{\sqrt{z^{2}+\alpha}}+C\\ & =\frac{1}{2}\left(\Log\left(1+\frac{z}{\sqrt{z^{2}+\alpha}}\right)-\Log\left(1-\frac{z}{\sqrt{z^{2}+\alpha}}\right)\right)+C \end{align*}

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分母に2乗のルートがある積分

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