$$$\frac{1}{\csc{\left(x \right)}}$$$の積分

$$$\int \frac{1}{\csc{\left(x \right)}}\, dx$$$ を求めよ。

被積分関数を正弦関数で書き換えよ:

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

正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:

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

したがって、

$$\int{\frac{1}{\csc{\left(x \right)}} d x} = - \cos{\left(x \right)}$$

積分定数を加える:

$$\int{\frac{1}{\csc{\left(x \right)}} d x} = - \cos{\left(x \right)}+C$$

$$$\int \frac{1}{\csc{\left(x \right)}}\, dx = - \cos{\left(x \right)} + C$$$A