$$$x^{4} - \frac{1}{4 x^{4}}$$$의 적분

$$$\int \left(x^{4} - \frac{1}{4 x^{4}}\right)\, dx$$$을(를) 구하시오.

각 항별로 적분하십시오:

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

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=4$$$에 적용합니다:

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

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

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

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=-4$$$에 적용합니다:

$$\frac{x^{5}}{5} - \frac{{\color{red}{\int{\frac{1}{x^{4}} d x}}}}{4}=\frac{x^{5}}{5} - \frac{{\color{red}{\int{x^{-4} d x}}}}{4}=\frac{x^{5}}{5} - \frac{{\color{red}{\frac{x^{-4 + 1}}{-4 + 1}}}}{4}=\frac{x^{5}}{5} - \frac{{\color{red}{\left(- \frac{x^{-3}}{3}\right)}}}{4}=\frac{x^{5}}{5} - \frac{{\color{red}{\left(- \frac{1}{3 x^{3}}\right)}}}{4}$$

따라서,

$$\int{\left(x^{4} - \frac{1}{4 x^{4}}\right)d x} = \frac{x^{5}}{5} + \frac{1}{12 x^{3}}$$

간단히 하시오:

$$\int{\left(x^{4} - \frac{1}{4 x^{4}}\right)d x} = \frac{12 x^{8} + 5}{60 x^{3}}$$

적분 상수를 추가하세요:

$$\int{\left(x^{4} - \frac{1}{4 x^{4}}\right)d x} = \frac{12 x^{8} + 5}{60 x^{3}}+C$$

$$$\int \left(x^{4} - \frac{1}{4 x^{4}}\right)\, dx = \frac{12 x^{8} + 5}{60 x^{3}} + C$$$A