$$$\left(x - 2\right)^{4} \left(x - 1\right)^{3}$$$의 적분

$$$\int \left(x - 2\right)^{4} \left(x - 1\right)^{3}\, dx$$$을(를) 구하시오.

$$$u=x - 2$$$라 하자.

그러면 $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$dx = du$$$임을 얻습니다.

적분은 다음과 같이 됩니다.

$${\color{red}{\int{\left(x - 2\right)^{4} \left(x - 1\right)^{3} d x}}} = {\color{red}{\int{u^{4} \left(u + 1\right)^{3} d u}}}$$

Expand the expression:

$${\color{red}{\int{u^{4} \left(u + 1\right)^{3} d u}}} = {\color{red}{\int{\left(u^{7} + 3 u^{6} + 3 u^{5} + u^{4}\right)d u}}}$$

각 항별로 적분하십시오:

$${\color{red}{\int{\left(u^{7} + 3 u^{6} + 3 u^{5} + u^{4}\right)d u}}} = {\color{red}{\left(\int{u^{4} d u} + \int{3 u^{5} d u} + \int{3 u^{6} d u} + \int{u^{7} d u}\right)}}$$

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

$$\int{3 u^{5} d u} + \int{3 u^{6} d u} + \int{u^{7} d u} + {\color{red}{\int{u^{4} d u}}}=\int{3 u^{5} d u} + \int{3 u^{6} d u} + \int{u^{7} d u} + {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=\int{3 u^{5} d u} + \int{3 u^{6} d u} + \int{u^{7} d u} + {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

멱법칙($$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$)을 $$$n=7$$$에 적용합니다:

$$\frac{u^{5}}{5} + \int{3 u^{5} d u} + \int{3 u^{6} d u} + {\color{red}{\int{u^{7} d u}}}=\frac{u^{5}}{5} + \int{3 u^{5} d u} + \int{3 u^{6} d u} + {\color{red}{\frac{u^{1 + 7}}{1 + 7}}}=\frac{u^{5}}{5} + \int{3 u^{5} d u} + \int{3 u^{6} d u} + {\color{red}{\left(\frac{u^{8}}{8}\right)}}$$

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:

$$\frac{u^{8}}{8} + \frac{u^{5}}{5} + \int{3 u^{6} d u} + {\color{red}{\int{3 u^{5} d u}}} = \frac{u^{8}}{8} + \frac{u^{5}}{5} + \int{3 u^{6} d u} + {\color{red}{\left(3 \int{u^{5} d u}\right)}}$$

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

$$\frac{u^{8}}{8} + \frac{u^{5}}{5} + \int{3 u^{6} d u} + 3 {\color{red}{\int{u^{5} d u}}}=\frac{u^{8}}{8} + \frac{u^{5}}{5} + \int{3 u^{6} d u} + 3 {\color{red}{\frac{u^{1 + 5}}{1 + 5}}}=\frac{u^{8}}{8} + \frac{u^{5}}{5} + \int{3 u^{6} d u} + 3 {\color{red}{\left(\frac{u^{6}}{6}\right)}}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=3$$$와 $$$f{\left(u \right)} = u^{6}$$$에 적용하세요:

$$\frac{u^{8}}{8} + \frac{u^{6}}{2} + \frac{u^{5}}{5} + {\color{red}{\int{3 u^{6} d u}}} = \frac{u^{8}}{8} + \frac{u^{6}}{2} + \frac{u^{5}}{5} + {\color{red}{\left(3 \int{u^{6} d u}\right)}}$$

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

$$\frac{u^{8}}{8} + \frac{u^{6}}{2} + \frac{u^{5}}{5} + 3 {\color{red}{\int{u^{6} d u}}}=\frac{u^{8}}{8} + \frac{u^{6}}{2} + \frac{u^{5}}{5} + 3 {\color{red}{\frac{u^{1 + 6}}{1 + 6}}}=\frac{u^{8}}{8} + \frac{u^{6}}{2} + \frac{u^{5}}{5} + 3 {\color{red}{\left(\frac{u^{7}}{7}\right)}}$$

다음 $$$u=x - 2$$$을 기억하라:

$$\frac{{\color{red}{u}}^{5}}{5} + \frac{{\color{red}{u}}^{6}}{2} + \frac{3 {\color{red}{u}}^{7}}{7} + \frac{{\color{red}{u}}^{8}}{8} = \frac{{\color{red}{\left(x - 2\right)}}^{5}}{5} + \frac{{\color{red}{\left(x - 2\right)}}^{6}}{2} + \frac{3 {\color{red}{\left(x - 2\right)}}^{7}}{7} + \frac{{\color{red}{\left(x - 2\right)}}^{8}}{8}$$

따라서,

$$\int{\left(x - 2\right)^{4} \left(x - 1\right)^{3} d x} = \frac{\left(x - 2\right)^{8}}{8} + \frac{3 \left(x - 2\right)^{7}}{7} + \frac{\left(x - 2\right)^{6}}{2} + \frac{\left(x - 2\right)^{5}}{5}$$

간단히 하시오:

$$\int{\left(x - 2\right)^{4} \left(x - 1\right)^{3} d x} = \frac{\left(x - 2\right)^{5} \left(140 x + 35 \left(x - 2\right)^{3} + 120 \left(x - 2\right)^{2} - 224\right)}{280}$$

적분 상수를 추가하세요:

$$\int{\left(x - 2\right)^{4} \left(x - 1\right)^{3} d x} = \frac{\left(x - 2\right)^{5} \left(140 x + 35 \left(x - 2\right)^{3} + 120 \left(x - 2\right)^{2} - 224\right)}{280}+C$$

$$$\int \left(x - 2\right)^{4} \left(x - 1\right)^{3}\, dx = \frac{\left(x - 2\right)^{5} \left(140 x + 35 \left(x - 2\right)^{3} + 120 \left(x - 2\right)^{2} - 224\right)}{280} + C$$$A