$$$\sin{\left(x \right)} \cos{\left(2 \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \sin{\left(x \right)} \cos{\left(2 \right)}\, dx$$$ 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、$$$c=\cos{\left(2 \right)}$$$ と $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\sin{\left(x \right)} \cos{\left(2 \right)} d x}}} = {\color{red}{\cos{\left(2 \right)} \int{\sin{\left(x \right)} d x}}}$$
正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:
$$\cos{\left(2 \right)} {\color{red}{\int{\sin{\left(x \right)} d x}}} = \cos{\left(2 \right)} {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
したがって、
$$\int{\sin{\left(x \right)} \cos{\left(2 \right)} d x} = - \cos{\left(2 \right)} \cos{\left(x \right)}$$
積分定数を加える:
$$\int{\sin{\left(x \right)} \cos{\left(2 \right)} d x} = - \cos{\left(2 \right)} \cos{\left(x \right)}+C$$
解答
$$$\int \sin{\left(x \right)} \cos{\left(2 \right)}\, dx = - \cos{\left(2 \right)} \cos{\left(x \right)} + C$$$A