$$$x^{9} \ln\left(x\right)$$$ 的积分

求$$$\int x^{9} \ln\left(x\right)\, dx$$$。

对于积分$$$\int{x^{9} \ln{\left(x \right)} d x}$$$，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 $$$\operatorname{dv}=x^{9} dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{x^{9} d x}=\frac{x^{10}}{10}$$$ (步骤见 »)。

积分变为

$${\color{red}{\int{x^{9} \ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{10}}{10}-\int{\frac{x^{10}}{10} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\frac{x^{10} \ln{\left(x \right)}}{10} - \int{\frac{x^{9}}{10} d x}\right)}}$$

对 $$$c=\frac{1}{10}$$$ 和 $$$f{\left(x \right)} = x^{9}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{x^{10} \ln{\left(x \right)}}{10} - {\color{red}{\int{\frac{x^{9}}{10} d x}}} = \frac{x^{10} \ln{\left(x \right)}}{10} - {\color{red}{\left(\frac{\int{x^{9} d x}}{10}\right)}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=9$$$：

$$\frac{x^{10} \ln{\left(x \right)}}{10} - \frac{{\color{red}{\int{x^{9} d x}}}}{10}=\frac{x^{10} \ln{\left(x \right)}}{10} - \frac{{\color{red}{\frac{x^{1 + 9}}{1 + 9}}}}{10}=\frac{x^{10} \ln{\left(x \right)}}{10} - \frac{{\color{red}{\left(\frac{x^{10}}{10}\right)}}}{10}$$

因此，

$$\int{x^{9} \ln{\left(x \right)} d x} = \frac{x^{10} \ln{\left(x \right)}}{10} - \frac{x^{10}}{100}$$

化简：

$$\int{x^{9} \ln{\left(x \right)} d x} = \frac{x^{10} \left(10 \ln{\left(x \right)} - 1\right)}{100}$$

加上积分常数：

$$\int{x^{9} \ln{\left(x \right)} d x} = \frac{x^{10} \left(10 \ln{\left(x \right)} - 1\right)}{100}+C$$

$$$\int x^{9} \ln\left(x\right)\, dx = \frac{x^{10} \left(10 \ln\left(x\right) - 1\right)}{100} + C$$$A