$$$x^{2} - \frac{12}{x^{31}}$$$의 적분

$$$\int \left(x^{2} - \frac{12}{x^{31}}\right)\, dx$$$을(를) 구하시오.

각 항별로 적분하십시오:

$${\color{red}{\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x}}} = {\color{red}{\left(- \int{\frac{12}{x^{31}} d x} + \int{x^{2} d x}\right)}}$$

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=2$$$에 적용합니다:

$$- \int{\frac{12}{x^{31}} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{\frac{12}{x^{31}} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{\frac{12}{x^{31}} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

상수배 법칙 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$을 $$$c=12$$$와 $$$f{\left(x \right)} = \frac{1}{x^{31}}$$$에 적용하세요:

$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{12}{x^{31}} d x}}} = \frac{x^{3}}{3} - {\color{red}{\left(12 \int{\frac{1}{x^{31}} d x}\right)}}$$

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=-31$$$에 적용합니다:

$$\frac{x^{3}}{3} - 12 {\color{red}{\int{\frac{1}{x^{31}} d x}}}=\frac{x^{3}}{3} - 12 {\color{red}{\int{x^{-31} d x}}}=\frac{x^{3}}{3} - 12 {\color{red}{\frac{x^{-31 + 1}}{-31 + 1}}}=\frac{x^{3}}{3} - 12 {\color{red}{\left(- \frac{x^{-30}}{30}\right)}}=\frac{x^{3}}{3} - 12 {\color{red}{\left(- \frac{1}{30 x^{30}}\right)}}$$

따라서,

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{x^{3}}{3} + \frac{2}{5 x^{30}}$$

간단히 하시오:

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{5 x^{33} + 6}{15 x^{30}}$$

적분 상수를 추가하세요:

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{5 x^{33} + 6}{15 x^{30}}+C$$

$$$\int \left(x^{2} - \frac{12}{x^{31}}\right)\, dx = \frac{5 x^{33} + 6}{15 x^{30}} + C$$$A