$$$- \sin{\left(x \right)}$$$の積分
入力内容
$$$\int \left(- \sin{\left(x \right)}\right)\, dx$$$ 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、$$$c=-1$$$ と $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\left(- \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\sin{\left(x \right)} d x}\right)}}$$
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、
$$\int{\left(- \sin{\left(x \right)}\right)d x} = \cos{\left(x \right)}$$
積分定数を加える:
$$\int{\left(- \sin{\left(x \right)}\right)d x} = \cos{\left(x \right)}+C$$
解答
$$$\int \left(- \sin{\left(x \right)}\right)\, dx = \cos{\left(x \right)} + C$$$A