$$$\frac{z \left(6 - 2 z\right)^{5}}{3}$$$의 적분

$$$\int \frac{z \left(6 - 2 z\right)^{5}}{3}\, dz$$$을(를) 구하시오.

피적분함수를 단순화하세요.:

$${\color{red}{\int{\frac{z \left(6 - 2 z\right)^{5}}{3} d z}}} = {\color{red}{\int{\frac{32 z \left(3 - z\right)^{5}}{3} d z}}}$$

상수배 법칙 $$$\int c f{\left(z \right)}\, dz = c \int f{\left(z \right)}\, dz$$$을 $$$c=\frac{32}{3}$$$와 $$$f{\left(z \right)} = z \left(3 - z\right)^{5}$$$에 적용하세요:

$${\color{red}{\int{\frac{32 z \left(3 - z\right)^{5}}{3} d z}}} = {\color{red}{\left(\frac{32 \int{z \left(3 - z\right)^{5} d z}}{3}\right)}}$$

$$$u=3 - z$$$라 하자.

그러면 $$$du=\left(3 - z\right)^{\prime }dz = - dz$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$dz = - du$$$임을 얻습니다.

따라서,

$$\frac{32 {\color{red}{\int{z \left(3 - z\right)^{5} d z}}}}{3} = \frac{32 {\color{red}{\int{u^{5} \left(u - 3\right) d u}}}}{3}$$

Expand the expression:

$$\frac{32 {\color{red}{\int{u^{5} \left(u - 3\right) d u}}}}{3} = \frac{32 {\color{red}{\int{\left(u^{6} - 3 u^{5}\right)d u}}}}{3}$$

각 항별로 적분하십시오:

$$\frac{32 {\color{red}{\int{\left(u^{6} - 3 u^{5}\right)d u}}}}{3} = \frac{32 {\color{red}{\left(- \int{3 u^{5} d u} + \int{u^{6} d u}\right)}}}{3}$$

멱법칙($$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=6$$$에 적용합니다:

$$- \frac{32 \int{3 u^{5} d u}}{3} + \frac{32 {\color{red}{\int{u^{6} d u}}}}{3}=- \frac{32 \int{3 u^{5} d u}}{3} + \frac{32 {\color{red}{\frac{u^{1 + 6}}{1 + 6}}}}{3}=- \frac{32 \int{3 u^{5} d u}}{3} + \frac{32 {\color{red}{\left(\frac{u^{7}}{7}\right)}}}{3}$$

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

$$\frac{32 u^{7}}{21} - \frac{32 {\color{red}{\int{3 u^{5} d u}}}}{3} = \frac{32 u^{7}}{21} - \frac{32 {\color{red}{\left(3 \int{u^{5} d u}\right)}}}{3}$$

멱법칙($$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=5$$$에 적용합니다:

$$\frac{32 u^{7}}{21} - 32 {\color{red}{\int{u^{5} d u}}}=\frac{32 u^{7}}{21} - 32 {\color{red}{\frac{u^{1 + 5}}{1 + 5}}}=\frac{32 u^{7}}{21} - 32 {\color{red}{\left(\frac{u^{6}}{6}\right)}}$$

다음 $$$u=3 - z$$$을 기억하라:

$$- \frac{16 {\color{red}{u}}^{6}}{3} + \frac{32 {\color{red}{u}}^{7}}{21} = - \frac{16 {\color{red}{\left(3 - z\right)}}^{6}}{3} + \frac{32 {\color{red}{\left(3 - z\right)}}^{7}}{21}$$

따라서,

$$\int{\frac{z \left(6 - 2 z\right)^{5}}{3} d z} = \frac{32 \left(3 - z\right)^{7}}{21} - \frac{16 \left(3 - z\right)^{6}}{3}$$

간단히 하시오:

$$\int{\frac{z \left(6 - 2 z\right)^{5}}{3} d z} = \frac{16 \left(- 2 z - 1\right) \left(z - 3\right)^{6}}{21}$$

적분 상수를 추가하세요:

$$\int{\frac{z \left(6 - 2 z\right)^{5}}{3} d z} = \frac{16 \left(- 2 z - 1\right) \left(z - 3\right)^{6}}{21}+C$$

$$$\int \frac{z \left(6 - 2 z\right)^{5}}{3}\, dz = \frac{16 \left(- 2 z - 1\right) \left(z - 3\right)^{6}}{21} + C$$$A