$$$\frac{x^{5} - 4}{x^{22}}$$$의 적분

$$$\int \frac{x^{5} - 4}{x^{22}}\, dx$$$을(를) 구하시오.

Expand the expression:

$${\color{red}{\int{\frac{x^{5} - 4}{x^{22}} d x}}} = {\color{red}{\int{\left(\frac{1}{x^{17}} - \frac{4}{x^{22}}\right)d x}}}$$

각 항별로 적분하십시오:

$${\color{red}{\int{\left(\frac{1}{x^{17}} - \frac{4}{x^{22}}\right)d x}}} = {\color{red}{\left(- \int{\frac{4}{x^{22}} d x} + \int{\frac{1}{x^{17}} d x}\right)}}$$

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=-17$$$에 적용합니다:

$$- \int{\frac{4}{x^{22}} d x} + {\color{red}{\int{\frac{1}{x^{17}} d x}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\int{x^{-17} d x}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\frac{x^{-17 + 1}}{-17 + 1}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\left(- \frac{x^{-16}}{16}\right)}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\left(- \frac{1}{16 x^{16}}\right)}}$$

상수배 법칙 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$을 $$$c=4$$$와 $$$f{\left(x \right)} = \frac{1}{x^{22}}$$$에 적용하세요:

$$- {\color{red}{\int{\frac{4}{x^{22}} d x}}} - \frac{1}{16 x^{16}} = - {\color{red}{\left(4 \int{\frac{1}{x^{22}} d x}\right)}} - \frac{1}{16 x^{16}}$$

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=-22$$$에 적용합니다:

$$- 4 {\color{red}{\int{\frac{1}{x^{22}} d x}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\int{x^{-22} d x}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\frac{x^{-22 + 1}}{-22 + 1}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\left(- \frac{x^{-21}}{21}\right)}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\left(- \frac{1}{21 x^{21}}\right)}} - \frac{1}{16 x^{16}}$$

따라서,

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = - \frac{1}{16 x^{16}} + \frac{4}{21 x^{21}}$$

간단히 하시오:

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = \frac{64 - 21 x^{5}}{336 x^{21}}$$

적분 상수를 추가하세요:

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = \frac{64 - 21 x^{5}}{336 x^{21}}+C$$

$$$\int \frac{x^{5} - 4}{x^{22}}\, dx = \frac{64 - 21 x^{5}}{336 x^{21}} + C$$$A