|
|
|
|
|
|
|
|
|
|
|
|
$$$\frac{\sin{\left(x \right)}}{2}$$$の積分
入力内容
$$$\int \frac{\sin{\left(x \right)}}{2}\, dx$$$ 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、$$$c=\frac{1}{2}$$$ と $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\frac{\sin{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{\int{\sin{\left(x \right)} d x}}{2}\right)}}$$
正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:
$$\frac{{\color{red}{\int{\sin{\left(x \right)} d x}}}}{2} = \frac{{\color{red}{\left(- \cos{\left(x \right)}\right)}}}{2}$$
したがって、
$$\int{\frac{\sin{\left(x \right)}}{2} d x} = - \frac{\cos{\left(x \right)}}{2}$$
積分定数を加える:
$$\int{\frac{\sin{\left(x \right)}}{2} d x} = - \frac{\cos{\left(x \right)}}{2}+C$$
解答
$$$\int \frac{\sin{\left(x \right)}}{2}\, dx = - \frac{\cos{\left(x \right)}}{2} + C$$$A