$$$a^{\frac{x}{b}}$$$ 對 $$$x$$$ 的積分

求$$$\int a^{\frac{x}{b}}\, dx$$$。

令 $$$u=\frac{x}{b}$$$。

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

所以，

$${\color{red}{\int{a^{\frac{x}{b}} d x}}} = {\color{red}{\int{a^{u} b d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=b$$$ 與 $$$f{\left(u \right)} = a^{u}$$$：

$${\color{red}{\int{a^{u} b d u}}} = {\color{red}{b \int{a^{u} d u}}}$$

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

$$b {\color{red}{\int{a^{u} d u}}} = b {\color{red}{\frac{a^{u}}{\ln{\left(a \right)}}}}$$

回顧一下 $$$u=\frac{x}{b}$$$：

$$\frac{b a^{{\color{red}{u}}}}{\ln{\left(a \right)}} = \frac{b a^{{\color{red}{\frac{x}{b}}}}}{\ln{\left(a \right)}}$$

因此，

$$\int{a^{\frac{x}{b}} d x} = \frac{a^{\frac{x}{b}} b}{\ln{\left(a \right)}}$$

加上積分常數：

$$\int{a^{\frac{x}{b}} d x} = \frac{a^{\frac{x}{b}} b}{\ln{\left(a \right)}}+C$$

$$$\int a^{\frac{x}{b}}\, dx = \frac{a^{\frac{x}{b}} b}{\ln\left(a\right)} + C$$$A