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$$$x^{4} - x^{3}$$$ 的積分
您的輸入
求$$$\int \left(x^{4} - x^{3}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(x^{4} - x^{3}\right)d x}}} = {\color{red}{\left(- \int{x^{3} d x} + \int{x^{4} d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=4$$$：
$$- \int{x^{3} d x} + {\color{red}{\int{x^{4} d x}}}=- \int{x^{3} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \int{x^{3} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=3$$$：
$$\frac{x^{5}}{5} - {\color{red}{\int{x^{3} d x}}}=\frac{x^{5}}{5} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{5}}{5} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
因此，
$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{5}}{5} - \frac{x^{4}}{4}$$
化簡：
$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{4} \left(4 x - 5\right)}{20}$$
加上積分常數：
$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{4} \left(4 x - 5\right)}{20}+C$$
答案
$$$\int \left(x^{4} - x^{3}\right)\, dx = \frac{x^{4} \left(4 x - 5\right)}{20} + C$$$A