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*}ページ情報
タイトル | 冪関数と指数関数の積の積分 |
URL | https://www.nomuramath.com/nmp735fv/ |
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逆関数の微分
\[
\frac{df^{\bullet}(x)}{dx}=\left(\frac{df(f^{\bullet}(x))}{df^{\bullet}(x)}\right)^{-1}
\]
部分積分と繰り返し部分積分
\[
\int f(x)g(x)dx=\sum_{k=0}^{n-1}\left(-1\right)^{k}f^{(-(k+1))}(x)g^{(k)}(x)+(-1)^{n}\int f^{(-n)}(x)g^{(n)}(x)dx
\]
微分・原始関数・定積分・不定積分の定義
\[
\frac{df(x)}{dx}=\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
\]
合成関数の微分
\[
\frac{df(g(x))}{dx}=f'(g(x))g'(x)
\]