$$$\frac{\theta \sin{\left(1 \right)}}{4}$$$ 的積分
您的輸入
求$$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta$$$。
解答
套用常數倍法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$，使用 $$$c=\frac{\sin{\left(1 \right)}}{4}$$$ 與 $$$f{\left(\theta \right)} = \theta$$$：
$${\color{red}{\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta}}} = {\color{red}{\left(\frac{\sin{\left(1 \right)} \int{\theta d \theta}}{4}\right)}}$$
套用冪次法則 $$$\int \theta^{n}\, d\theta = \frac{\theta^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：
$$\frac{\sin{\left(1 \right)} {\color{red}{\int{\theta d \theta}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\frac{\theta^{1 + 1}}{1 + 1}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\left(\frac{\theta^{2}}{2}\right)}}}{4}$$
因此，
$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}$$
加上積分常數：
$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}+C$$
答案
$$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta = \frac{\theta^{2} \sin{\left(1 \right)}}{8} + C$$$A