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$$$\frac{\csc^{2}{\left(x \right)}}{9}$$$ 的积分
您的输入
求$$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx$$$。
解答
对 $$$c=\frac{1}{9}$$$ 和 $$$f{\left(x \right)} = \csc^{2}{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\frac{\csc^{2}{\left(x \right)}}{9} d x}}} = {\color{red}{\left(\frac{\int{\csc^{2}{\left(x \right)} d x}}{9}\right)}}$$
$$$\csc^{2}{\left(x \right)}$$$ 的积分为 $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:
$$\frac{{\color{red}{\int{\csc^{2}{\left(x \right)} d x}}}}{9} = \frac{{\color{red}{\left(- \cot{\left(x \right)}\right)}}}{9}$$
因此，
$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}$$
加上积分常数：
$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}+C$$
答案
$$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx = - \frac{\cot{\left(x \right)}}{9} + C$$$A