$$$x e^{37} \ln\left(x\right)$$$ 的积分

求$$$\int x e^{37} \ln\left(x\right)\, dx$$$。

对 $$$c=e^{37}$$$ 和 $$$f{\left(x \right)} = x \ln{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{x e^{37} \ln{\left(x \right)} d x}}} = {\color{red}{e^{37} \int{x \ln{\left(x \right)} d x}}}$$

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

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (步骤见 »)。

因此，

$$e^{37} {\color{red}{\int{x \ln{\left(x \right)} d x}}}=e^{37} {\color{red}{\left(\ln{\left(x \right)} \cdot \frac{x^{2}}{2}-\int{\frac{x^{2}}{2} \cdot \frac{1}{x} d x}\right)}}=e^{37} {\color{red}{\left(\frac{x^{2} \ln{\left(x \right)}}{2} - \int{\frac{x}{2} d x}\right)}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = x$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$e^{37} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - {\color{red}{\int{\frac{x}{2} d x}}}\right) = e^{37} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - {\color{red}{\left(\frac{\int{x d x}}{2}\right)}}\right)$$

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

$$e^{37} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\int{x d x}}}}{2}\right)=e^{37} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{2}\right)=e^{37} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{2}\right)$$

因此，

$$\int{x e^{37} \ln{\left(x \right)} d x} = \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4}\right) e^{37}$$

化简：

$$\int{x e^{37} \ln{\left(x \right)} d x} = \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right) e^{37}}{4}$$

加上积分常数：

$$\int{x e^{37} \ln{\left(x \right)} d x} = \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right) e^{37}}{4}+C$$

$$$\int x e^{37} \ln\left(x\right)\, dx = \frac{x^{2} \left(2 \ln\left(x\right) - 1\right) e^{37}}{4} + C$$$A