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

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

令 $$$u=\sin{\left(x \right)}$$$。

則 $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(x \right)} dx = du$$$。

因此，

$${\color{red}{\int{\frac{\cot{\left(x \right)}}{\ln{\left(\sin{\left(x \right)} \right)}} d x}}} = {\color{red}{\int{\frac{1}{u \ln{\left(u \right)}} d u}}}$$

令 $$$v=\ln{\left(u \right)}$$$。

則 $$$dv=\left(\ln{\left(u \right)}\right)^{\prime }du = \frac{du}{u}$$$ (步驟見»)，並可得 $$$\frac{du}{u} = dv$$$。

該積分可改寫為

$${\color{red}{\int{\frac{1}{u \ln{\left(u \right)}} d u}}} = {\color{red}{\int{\frac{1}{v} d v}}}$$

$$$\frac{1}{v}$$$ 的積分是 $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$：

$${\color{red}{\int{\frac{1}{v} d v}}} = {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$

回顧一下 $$$v=\ln{\left(u \right)}$$$：

$$\ln{\left(\left|{{\color{red}{v}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\ln{\left(u \right)}}}}\right| \right)}$$

回顧一下 $$$u=\sin{\left(x \right)}$$$：

$$\ln{\left(\left|{\ln{\left({\color{red}{u}} \right)}}\right| \right)} = \ln{\left(\left|{\ln{\left({\color{red}{\sin{\left(x \right)}}} \right)}}\right| \right)}$$

因此，

$$\int{\frac{\cot{\left(x \right)}}{\ln{\left(\sin{\left(x \right)} \right)}} d x} = \ln{\left(\left|{\ln{\left(\sin{\left(x \right)} \right)}}\right| \right)}$$

加上積分常數：

$$\int{\frac{\cot{\left(x \right)}}{\ln{\left(\sin{\left(x \right)} \right)}} d x} = \ln{\left(\left|{\ln{\left(\sin{\left(x \right)} \right)}}\right| \right)}+C$$

$$$\int \frac{\cot{\left(x \right)}}{\ln\left(\sin{\left(x \right)}\right)}\, dx = \ln\left(\left|{\ln\left(\sin{\left(x \right)}\right)}\right|\right) + C$$$A