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$$$x^{2} - \frac{1}{x^{24}}$$$ 的積分
您的輸入
求$$$\int \left(x^{2} - \frac{1}{x^{24}}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(x^{2} - \frac{1}{x^{24}}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{x^{24}} d x} + \int{x^{2} d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：
$$- \int{\frac{1}{x^{24}} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{\frac{1}{x^{24}} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{\frac{1}{x^{24}} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-24$$$：
$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{1}{x^{24}} d x}}}=\frac{x^{3}}{3} - {\color{red}{\int{x^{-24} d x}}}=\frac{x^{3}}{3} - {\color{red}{\frac{x^{-24 + 1}}{-24 + 1}}}=\frac{x^{3}}{3} - {\color{red}{\left(- \frac{x^{-23}}{23}\right)}}=\frac{x^{3}}{3} - {\color{red}{\left(- \frac{1}{23 x^{23}}\right)}}$$
因此，
$$\int{\left(x^{2} - \frac{1}{x^{24}}\right)d x} = \frac{x^{3}}{3} + \frac{1}{23 x^{23}}$$
化簡：
$$\int{\left(x^{2} - \frac{1}{x^{24}}\right)d x} = \frac{23 x^{26} + 3}{69 x^{23}}$$
加上積分常數：
$$\int{\left(x^{2} - \frac{1}{x^{24}}\right)d x} = \frac{23 x^{26} + 3}{69 x^{23}}+C$$
答案
$$$\int \left(x^{2} - \frac{1}{x^{24}}\right)\, dx = \frac{23 x^{26} + 3}{69 x^{23}} + C$$$A