冪関数と指数関数の積の積分
冪関数と指数関数の積の積分
(1)
\[ \int z^{\alpha}e^{\beta z}dz=\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C \](2)
\[ \int z^{\alpha}\beta^{z}dz=\frac{z^{\alpha}}{\Log\beta\left(-z\Log\beta\right)^{\alpha}}\Gamma\left(\alpha+1,-z\Log\beta\right)+C \]-
\(\Gamma\left(x,y\right)\)は第2種不完全ガンマ関数(1)
\begin{align*} \int z^{\alpha}e^{\beta z}dz & =\frac{\left(-\beta\right)^{\alpha}z^{\alpha}}{\left(-\beta z\right)^{\alpha}}\int\frac{\left(-\beta z\right)^{\alpha}}{\left(-\beta\right)^{\alpha+1}}e^{\beta z}d\left(-\beta z\right)\\ & =\frac{z^{\alpha}}{\left(-\beta\right)\left(-\beta z\right)^{\alpha}}\int^{-\beta z}z^{\alpha}e^{-z}dz\\ & =\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C \end{align*}(2)
\begin{align*} \int z^{\alpha}\beta^{z}dz & =\int z^{\alpha}e^{z\Log\beta}dz\\ & =\left[\int z^{\alpha}e^{\beta z}dz\right]_{\beta\rightarrow\Log\beta}\\ & =\left[\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C\right]_{\beta\rightarrow\Log\beta}\\ & =\frac{z^{\alpha}}{\Log\beta\left(-z\Log\beta\right)^{\alpha}}\Gamma\left(\alpha+1,-z\Log\beta\right)+C \end{align*}
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\]
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\int_{-c}^{c}\frac{f_{e}\left(x\right)}{1+a^{x}}dx=\int_{0}^{c}f_{e}\left(x\right)dx
\]
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\[
\int f'(g(x))g'(x)dx=f(g(x))
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