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$$$x^{6} - \frac{1}{x^{21}}$$$の積分
入力内容
$$$\int \left(x^{6} - \frac{1}{x^{21}}\right)\, dx$$$ を求めよ。
解答
項別に積分せよ:
$${\color{red}{\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{x^{21}} d x} + \int{x^{6} d x}\right)}}$$
$$$n=6$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$- \int{\frac{1}{x^{21}} d x} + {\color{red}{\int{x^{6} d x}}}=- \int{\frac{1}{x^{21}} d x} + {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- \int{\frac{1}{x^{21}} d x} + {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$
$$$n=-21$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:
$$\frac{x^{7}}{7} - {\color{red}{\int{\frac{1}{x^{21}} d x}}}=\frac{x^{7}}{7} - {\color{red}{\int{x^{-21} d x}}}=\frac{x^{7}}{7} - {\color{red}{\frac{x^{-21 + 1}}{-21 + 1}}}=\frac{x^{7}}{7} - {\color{red}{\left(- \frac{x^{-20}}{20}\right)}}=\frac{x^{7}}{7} - {\color{red}{\left(- \frac{1}{20 x^{20}}\right)}}$$
したがって、
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{x^{7}}{7} + \frac{1}{20 x^{20}}$$
簡単化せよ:
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{27} + 7}{140 x^{20}}$$
積分定数を加える:
$$\int{\left(x^{6} - \frac{1}{x^{21}}\right)d x} = \frac{20 x^{27} + 7}{140 x^{20}}+C$$
解答
$$$\int \left(x^{6} - \frac{1}{x^{21}}\right)\, dx = \frac{20 x^{27} + 7}{140 x^{20}} + C$$$A