$$$\cot{\left(x + \frac{\pi}{4} \right)}$$$ 的積分
您的輸入
求$$$\int \cot{\left(x + \frac{\pi}{4} \right)}\, dx$$$。
解答
令 $$$u=x + \frac{\pi}{4}$$$。
則 $$$du=\left(x + \frac{\pi}{4}\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。
該積分變為
$${\color{red}{\int{\cot{\left(x + \frac{\pi}{4} \right)} d x}}} = {\color{red}{\int{\cot{\left(u \right)} d u}}}$$
將餘切改寫為 $$$\cot\left( u \right)=\frac{\cos\left( u \right)}{\sin\left( u \right)}$$$:
$${\color{red}{\int{\cot{\left(u \right)} d u}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}$$
令 $$$v=\sin{\left(u \right)}$$$。
則 $$$dv=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du$$$ (步驟見»)，並可得 $$$\cos{\left(u \right)} du = dv$$$。
該積分可改寫為
$${\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}} = {\color{red}{\int{\frac{1}{v} d v}}}$$
$$$\frac{1}{v}$$$ 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 $$$v=\sin{\left(u \right)}$$$：
$$\ln{\left(\left|{{\color{red}{v}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}$$
回顧一下 $$$u=x + \frac{\pi}{4}$$$：
$$\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)} = \ln{\left(\left|{\sin{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)}}\right| \right)}$$
因此，
$$\int{\cot{\left(x + \frac{\pi}{4} \right)} d x} = \ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}$$
加上積分常數：
$$\int{\cot{\left(x + \frac{\pi}{4} \right)} d x} = \ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}+C$$
答案
$$$\int \cot{\left(x + \frac{\pi}{4} \right)}\, dx = \ln\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right|\right) + C$$$A