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$$$x^{6} - \frac{1}{x^{21}}$$$ 的积分
您的输入
求$$$\int \left(x^{6} - \frac{1}{x^{21}}\right)\, dx$$$。
解答
逐项积分：
$${\color{red}{\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{x^{21}} d x} + \int{x^{6} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=6$$$：
$$- \int{\frac{1}{x^{21}} d x} + {\color{red}{\int{x^{6} d x}}}=- \int{\frac{1}{x^{21}} d x} + {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- \int{\frac{1}{x^{21}} d x} + {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-21$$$：
$$\frac{x^{7}}{7} - {\color{red}{\int{\frac{1}{x^{21}} d x}}}=\frac{x^{7}}{7} - {\color{red}{\int{x^{-21} d x}}}=\frac{x^{7}}{7} - {\color{red}{\frac{x^{-21 + 1}}{-21 + 1}}}=\frac{x^{7}}{7} - {\color{red}{\left(- \frac{x^{-20}}{20}\right)}}=\frac{x^{7}}{7} - {\color{red}{\left(- \frac{1}{20 x^{20}}\right)}}$$
因此，
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{x^{7}}{7} + \frac{1}{20 x^{20}}$$
化简：
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{27} + 7}{140 x^{20}}$$
加上积分常数：
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{27} + 7}{140 x^{20}}+C$$
答案
$$$\int \left(x^{6} - \frac{1}{x^{21}}\right)\, dx = \frac{20 x^{27} + 7}{140 x^{20}} + C$$$A