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

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

已將輸入重寫為：$$$\int{\left(- \frac{2^{- \frac{3 x^{2}}{5}} x}{5}\right)d x}=\int{\left(- \frac{x \left(\frac{2^{\frac{2}{5}}}{2}\right)^{x^{2}}}{5}\right)d x}$$$。

令 $$$u=x^{2}$$$。

則 $$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$ (步驟見»)，並可得 $$$x dx = \frac{du}{2}$$$。

該積分可改寫為

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=- \frac{1}{10}$$$ 與 $$$f{\left(u \right)} = \left(\frac{2^{\frac{2}{5}}}{2}\right)^{u}$$$：

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

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

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

回顧一下 $$$u=x^{2}$$$：

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

因此，

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

化簡：

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

加上積分常數：

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

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