$$$- \frac{2^{- \frac{3 x^{2}}{5}} x}{5}$$$の積分

$$$\int \left(- \frac{2^{- \frac{3 x^{2}}{5}} x}{5}\right)\, dx$$$ を求めよ。

入力は次のように書き換えられます: $$$\int{\left(- \frac{2^{- \frac{3 x^{2}}{5}} x}{5}\right)d x}=\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x}$$$。

$$$u=x^{2}$$$ とする。

すると $$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$（手順は»で確認できます）、$$$x dx = \frac{du}{2}$$$ となります。

したがって、

$${\color{red}{\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x}}} = {\color{red}{\int{\left(- \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{u}}{10}\right)d u}}}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=- \frac{1}{10}$$$ と $$$f{\left(u \right)} = \left(\frac{2^{\frac{2}{5}}}{2}\right)^{u}$$$ に対して適用する:

$${\color{red}{\int{\left(- \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{u}}{10}\right)d u}}} = {\color{red}{\left(- \frac{\int{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{u} d u}}{10}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{2^{\frac{2}{5}}}{2}$$$:

$$- \frac{{\color{red}{\int{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{u} d u}}}}{10} = - \frac{{\color{red}{\frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{u}}{\ln{\left(\frac{2^{\frac{2}{5}}}{2} \right)}}}}}{10}$$

次のことを思い出してください $$$u=x^{2}$$$:

$$- \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{{\color{red}{u}}}}{10 \ln{\left(\frac{2^{\frac{2}{5}}}{2} \right)}} = - \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{{\color{red}{x^{2}}}}}{10 \ln{\left(\frac{2^{\frac{2}{5}}}{2} \right)}}$$

したがって、

$$\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x} = - \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{10 \ln{\left(\frac{2^{\frac{2}{5}}}{2} \right)}}$$

簡単化せよ:

$$\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x} = \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{6 \ln{\left(2 \right)}}$$

積分定数を加える:

$$\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x} = \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{6 \ln{\left(2 \right)}}+C$$

$$$\int \left(- \frac{2^{- \frac{3 x^{2}}{5}} x}{5}\right)\, dx = \frac{\left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{6 \ln\left(2\right)} + C$$$A