$$$6 \cot{\left(x \right)} \csc{\left(x \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=6$$$ と $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(6 \int{\cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$
$$$\cot{\left(x \right)} \csc{\left(x \right)}$$$ の不定積分は $$$\int{\cot{\left(x \right)} \csc{\left(x \right)} d x} = - \csc{\left(x \right)}$$$ です:
$$6 {\color{red}{\int{\cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 6 {\color{red}{\left(- \csc{\left(x \right)}\right)}}$$
したがって、
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}$$
積分定数を加える:
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}+C$$
解答
$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx = - 6 \csc{\left(x \right)} + C$$$A