$$$5 x^{3} - x^{2}$$$의 적분

$$$\int \left(5 x^{3} - x^{2}\right)\, dx$$$을(를) 구하시오.

각 항별로 적분하십시오:

$${\color{red}{\int{\left(5 x^{3} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{5 x^{3} d x}\right)}}$$

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

$$\int{5 x^{3} d x} - {\color{red}{\int{x^{2} d x}}}=\int{5 x^{3} d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{5 x^{3} d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

상수배 법칙 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$을 $$$c=5$$$와 $$$f{\left(x \right)} = x^{3}$$$에 적용하세요:

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

멱법칙($$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=3$$$에 적용합니다:

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

따라서,

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{5 x^{4}}{4} - \frac{x^{3}}{3}$$

간단히 하시오:

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(15 x - 4\right)}{12}$$

적분 상수를 추가하세요:

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(15 x - 4\right)}{12}+C$$

$$$\int \left(5 x^{3} - x^{2}\right)\, dx = \frac{x^{3} \left(15 x - 4\right)}{12} + C$$$A