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