$$$\frac{\theta \sin{\left(1 \right)}}{4}$$$の積分
入力内容
$$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ を、$$$c=\frac{\sin{\left(1 \right)}}{4}$$$ と $$$f{\left(\theta \right)} = \theta$$$ に対して適用する:
$${\color{red}{\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta}}} = {\color{red}{\left(\frac{\sin{\left(1 \right)} \int{\theta d \theta}}{4}\right)}}$$
$$$n=1$$$ を用いて、べき乗の法則 $$$\int \theta^{n}\, d\theta = \frac{\theta^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{\sin{\left(1 \right)} {\color{red}{\int{\theta d \theta}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\frac{\theta^{1 + 1}}{1 + 1}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\left(\frac{\theta^{2}}{2}\right)}}}{4}$$
したがって、
$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}$$
積分定数を加える:
$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}+C$$
解答
$$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta = \frac{\theta^{2} \sin{\left(1 \right)}}{8} + C$$$A