$$$\ln\left(x^{7}\right) - \ln\left(y^{3}\right)$$$ 关于$$$x$$$的积分

求$$$\int \left(7 \ln\left(x\right) - 3 \ln\left(y\right)\right)\, dx$$$。

输入已重写为：$$$\int{\left(\ln{\left(x^{7} \right)} - \ln{\left(y^{3} \right)}\right)d x}=\int{\left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)}\right)d x}$$$。

逐项积分：

$${\color{red}{\int{\left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)}\right)d x}}} = {\color{red}{\left(\int{7 \ln{\left(x \right)} d x} - \int{3 \ln{\left(y \right)} d x}\right)}}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=3 \ln{\left(y \right)}$$$：

$$\int{7 \ln{\left(x \right)} d x} - {\color{red}{\int{3 \ln{\left(y \right)} d x}}} = \int{7 \ln{\left(x \right)} d x} - {\color{red}{\left(3 x \ln{\left(y \right)}\right)}}$$

对 $$$c=7$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- 3 x \ln{\left(y \right)} + {\color{red}{\int{7 \ln{\left(x \right)} d x}}} = - 3 x \ln{\left(y \right)} + {\color{red}{\left(7 \int{\ln{\left(x \right)} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

积分变为

$$- 3 x \ln{\left(y \right)} + 7 {\color{red}{\int{\ln{\left(x \right)} d x}}}=- 3 x \ln{\left(y \right)} + 7 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=- 3 x \ln{\left(y \right)} + 7 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

$$7 x \ln{\left(x \right)} - 3 x \ln{\left(y \right)} - 7 {\color{red}{\int{1 d x}}} = 7 x \ln{\left(x \right)} - 3 x \ln{\left(y \right)} - 7 {\color{red}{x}}$$

因此，

$$\int{\left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)}\right)d x} = 7 x \ln{\left(x \right)} - 3 x \ln{\left(y \right)} - 7 x$$

化简：

$$\int{\left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)}\right)d x} = x \left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)} - 7\right)$$

加上积分常数：

$$\int{\left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)}\right)d x} = x \left(7 \ln{\left(x \right)} - 3 \ln{\left(y \right)} - 7\right)+C$$

$$$\int \left(7 \ln\left(x\right) - 3 \ln\left(y\right)\right)\, dx = x \left(7 \ln\left(x\right) - 3 \ln\left(y\right) - 7\right) + C$$$A