$$$\sin{\left(\ln\left(x\right) \right)}$$$ 的積分

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

對於積分 $$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\sin{\left(\ln{\left(x \right)} \right)}$$$ 與 $$$\operatorname{dv}=dx$$$。

則 $$$\operatorname{du}=\left(\sin{\left(\ln{\left(x \right)} \right)}\right)^{\prime }dx=\frac{\cos{\left(\ln{\left(x \right)} \right)}}{x} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

因此，

$${\color{red}{\int{\sin{\left(\ln{\left(x \right)} \right)} d x}}}={\color{red}{\left(\sin{\left(\ln{\left(x \right)} \right)} \cdot x-\int{x \cdot \frac{\cos{\left(\ln{\left(x \right)} \right)}}{x} d x}\right)}}={\color{red}{\left(x \sin{\left(\ln{\left(x \right)} \right)} - \int{\cos{\left(\ln{\left(x \right)} \right)} d x}\right)}}$$

對於積分 $$$\int{\cos{\left(\ln{\left(x \right)} \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\cos{\left(\ln{\left(x \right)} \right)}$$$ 與 $$$\operatorname{dv}=dx$$$。

則 $$$\operatorname{du}=\left(\cos{\left(\ln{\left(x \right)} \right)}\right)^{\prime }dx=- \frac{\sin{\left(\ln{\left(x \right)} \right)}}{x} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

因此，

$$x \sin{\left(\ln{\left(x \right)} \right)} - {\color{red}{\int{\cos{\left(\ln{\left(x \right)} \right)} d x}}}=x \sin{\left(\ln{\left(x \right)} \right)} - {\color{red}{\left(\cos{\left(\ln{\left(x \right)} \right)} \cdot x-\int{x \cdot \left(- \frac{\sin{\left(\ln{\left(x \right)} \right)}}{x}\right) d x}\right)}}=x \sin{\left(\ln{\left(x \right)} \right)} - {\color{red}{\left(x \cos{\left(\ln{\left(x \right)} \right)} - \int{\left(- \sin{\left(\ln{\left(x \right)} \right)}\right)d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-1$$$ 與 $$$f{\left(x \right)} = \sin{\left(\ln{\left(x \right)} \right)}$$$：

$$x \sin{\left(\ln{\left(x \right)} \right)} - x \cos{\left(\ln{\left(x \right)} \right)} + {\color{red}{\int{\left(- \sin{\left(\ln{\left(x \right)} \right)}\right)d x}}} = x \sin{\left(\ln{\left(x \right)} \right)} - x \cos{\left(\ln{\left(x \right)} \right)} + {\color{red}{\left(- \int{\sin{\left(\ln{\left(x \right)} \right)} d x}\right)}}$$

我們得到了先前見過的一個積分。

因此，我們得到關於該積分的如下簡單等式：

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = x \sin{\left(\ln{\left(x \right)} \right)} - x \cos{\left(\ln{\left(x \right)} \right)} - \int{\sin{\left(\ln{\left(x \right)} \right)} d x}$$

求解後，可得

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = \frac{x \left(\sin{\left(\ln{\left(x \right)} \right)} - \cos{\left(\ln{\left(x \right)} \right)}\right)}{2}$$

因此，

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = \frac{x \left(\sin{\left(\ln{\left(x \right)} \right)} - \cos{\left(\ln{\left(x \right)} \right)}\right)}{2}$$

化簡：

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = - \frac{\sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}$$

加上積分常數：

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = - \frac{\sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+C$$

$$$\int \sin{\left(\ln\left(x\right) \right)}\, dx = - \frac{\sqrt{2} x \cos{\left(\ln\left(x\right) + \frac{\pi}{4} \right)}}{2} + C$$$A