$$$\ln\left(x\right) \cosh{\left(x \right)}$$$ 的积分
您的输入
求$$$\int \ln\left(x\right) \cosh{\left(x \right)}\, dx$$$。
解答
对于积分$$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x}$$$，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 $$$\operatorname{dv}=\cosh{\left(x \right)} dx$$$。
则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\cosh{\left(x \right)} d x}=\sinh{\left(x \right)}$$$ (步骤见 »)。
因此，
$${\color{red}{\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \sinh{\left(x \right)}-\int{\sinh{\left(x \right)} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\ln{\left(x \right)} \sinh{\left(x \right)} - \int{\frac{\sinh{\left(x \right)}}{x} d x}\right)}}$$
该积分（双曲正弦积分）没有闭式表达式：
$$\ln{\left(x \right)} \sinh{\left(x \right)} - {\color{red}{\int{\frac{\sinh{\left(x \right)}}{x} d x}}} = \ln{\left(x \right)} \sinh{\left(x \right)} - {\color{red}{\operatorname{Shi}{\left(x \right)}}}$$
因此，
$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x} = \ln{\left(x \right)} \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}$$
加上积分常数：
$$\int{\ln{\left(x \right)} \cosh{\left(x \right)} d x} = \ln{\left(x \right)} \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}+C$$
答案
$$$\int \ln\left(x\right) \cosh{\left(x \right)}\, dx = \left(\ln\left(x\right) \sinh{\left(x \right)} - \operatorname{Shi}{\left(x \right)}\right) + C$$$A