$$$x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)}\, dx$$$ 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、$$$c=\cot{\left(6 \right)} \csc{\left(4 \right)}$$$ と $$$f{\left(x \right)} = x^{2}$$$ に対して適用する:
$${\color{red}{\int{x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)} d x}}} = {\color{red}{\cot{\left(6 \right)} \csc{\left(4 \right)} \int{x^{2} d x}}}$$
$$$n=2$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\cot{\left(6 \right)} \csc{\left(4 \right)} {\color{red}{\int{x^{2} d x}}}=\cot{\left(6 \right)} \csc{\left(4 \right)} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\cot{\left(6 \right)} \csc{\left(4 \right)} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
したがって、
$$\int{x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)} d x} = \frac{x^{3} \cot{\left(6 \right)} \csc{\left(4 \right)}}{3}$$
積分定数を加える:
$$\int{x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)} d x} = \frac{x^{3} \cot{\left(6 \right)} \csc{\left(4 \right)}}{3}+C$$
解答
$$$\int x^{2} \cot{\left(6 \right)} \csc{\left(4 \right)}\, dx = \frac{x^{3} \cot{\left(6 \right)} \csc{\left(4 \right)}}{3} + C$$$A