$$$\frac{1}{\ln\left(x^{2}\right)}$$$ 的积分
您的输入
求$$$\int \frac{1}{\ln\left(x^{2}\right)}\, dx$$$。
解答
输入已重写为：$$$\int{\frac{1}{\ln{\left(x^{2} \right)}} d x}=\int{\frac{1}{2 \ln{\left(x \right)}} d x}$$$。
对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \frac{1}{\ln{\left(x \right)}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\frac{1}{2 \ln{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\ln{\left(x \right)}} d x}}{2}\right)}}$$
该积分（对数积分）没有闭式表达式：
$$\frac{{\color{red}{\int{\frac{1}{\ln{\left(x \right)}} d x}}}}{2} = \frac{{\color{red}{\operatorname{li}{\left(x \right)}}}}{2}$$
因此，
$$\int{\frac{1}{2 \ln{\left(x \right)}} d x} = \frac{\operatorname{li}{\left(x \right)}}{2}$$
加上积分常数：
$$\int{\frac{1}{2 \ln{\left(x \right)}} d x} = \frac{\operatorname{li}{\left(x \right)}}{2}+C$$
答案
$$$\int \frac{1}{\ln\left(x^{2}\right)}\, dx = \frac{\operatorname{li}{\left(x \right)}}{2} + C$$$A