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$$$- \frac{4}{x} + \frac{3}{x^{21}}$$$ 的积分
您的输入
求$$$\int \left(- \frac{4}{x} + \frac{3}{x^{21}}\right)\, dx$$$。
解答
逐项积分：
$${\color{red}{\int{\left(- \frac{4}{x} + \frac{3}{x^{21}}\right)d x}}} = {\color{red}{\left(\int{\frac{3}{x^{21}} d x} - \int{\frac{4}{x} d x}\right)}}$$
对 $$$c=4$$$ 和 $$$f{\left(x \right)} = \frac{1}{x}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$\int{\frac{3}{x^{21}} d x} - {\color{red}{\int{\frac{4}{x} d x}}} = \int{\frac{3}{x^{21}} d x} - {\color{red}{\left(4 \int{\frac{1}{x} d x}\right)}}$$
$$$\frac{1}{x}$$$ 的积分为 $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$$\int{\frac{3}{x^{21}} d x} - 4 {\color{red}{\int{\frac{1}{x} d x}}} = \int{\frac{3}{x^{21}} d x} - 4 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{21}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$- 4 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{\frac{3}{x^{21}} d x}}} = - 4 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(3 \int{\frac{1}{x^{21}} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-21$$$：
$$- 4 \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\int{\frac{1}{x^{21}} d x}}}=- 4 \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\int{x^{-21} d x}}}=- 4 \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\frac{x^{-21 + 1}}{-21 + 1}}}=- 4 \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\left(- \frac{x^{-20}}{20}\right)}}=- 4 \ln{\left(\left|{x}\right| \right)} + 3 {\color{red}{\left(- \frac{1}{20 x^{20}}\right)}}$$
因此，
$$\int{\left(- \frac{4}{x} + \frac{3}{x^{21}}\right)d x} = - 4 \ln{\left(\left|{x}\right| \right)} - \frac{3}{20 x^{20}}$$
加上积分常数：
$$\int{\left(- \frac{4}{x} + \frac{3}{x^{21}}\right)d x} = - 4 \ln{\left(\left|{x}\right| \right)} - \frac{3}{20 x^{20}}+C$$
答案
$$$\int \left(- \frac{4}{x} + \frac{3}{x^{21}}\right)\, dx = \left(- 4 \ln\left(\left|{x}\right|\right) - \frac{3}{20 x^{20}}\right) + C$$$A