$$$6 \cot{\left(x \right)} \csc{\left(x \right)}$$$ 的積分
您的輸入
求$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=6$$$ 與 $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$：
$${\color{red}{\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(6 \int{\cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$
$$$\cot{\left(x \right)} \csc{\left(x \right)}$$$ 的積分是 $$$\int{\cot{\left(x \right)} \csc{\left(x \right)} d x} = - \csc{\left(x \right)}$$$：
$$6 {\color{red}{\int{\cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 6 {\color{red}{\left(- \csc{\left(x \right)}\right)}}$$
因此，
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}$$
加上積分常數：
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}+C$$
答案
$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx = - 6 \csc{\left(x \right)} + C$$$A