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