$$$a^{x} \ln\left(a\right)$$$ 关于$$$x$$$的积分

求$$$\int a^{x} \ln\left(a\right)\, dx$$$。

对 $$$c=\ln{\left(a \right)}$$$ 和 $$$f{\left(x \right)} = a^{x}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

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

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

因此，

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

加上积分常数：

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

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