$$$x^{2} \operatorname{atan}{\left(\sqrt{x} \right)}$$$의 적분

$$$\int x^{2} \operatorname{atan}{\left(\sqrt{x} \right)}\, dx$$$을(를) 구하시오.

적분 $$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x}$$$에 대해서는 부분적분법 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$을 사용하십시오.

$$$\operatorname{u}=\operatorname{atan}{\left(\sqrt{x} \right)}$$$와 $$$\operatorname{dv}=x^{2} dx$$$라고 하자.

그러면 $$$\operatorname{du}=\left(\operatorname{atan}{\left(\sqrt{x} \right)}\right)^{\prime }dx=\frac{1}{2 \sqrt{x} \left(x + 1\right)} dx$$$ (»에서 풀이 과정을 볼 수 있음) 및 $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$ (»에서 풀이 과정을 볼 수 있음).

적분은 다음과 같이 다시 쓸 수 있습니다.

$${\color{red}{\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x}}}={\color{red}{\left(\operatorname{atan}{\left(\sqrt{x} \right)} \cdot \frac{x^{3}}{3}-\int{\frac{x^{3}}{3} \cdot \frac{1}{2 \sqrt{x} \left(x + 1\right)} d x}\right)}}={\color{red}{\left(\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \int{\frac{x^{\frac{5}{2}}}{6 x + 6} d x}\right)}}$$

피적분함수를 단순화하세요.:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 x + 6} d x}}} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 \left(x + 1\right)} d x}}}$$

상수배 법칙 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$을 $$$c=\frac{1}{6}$$$와 $$$f{\left(x \right)} = \frac{x^{\frac{5}{2}}}{x + 1}$$$에 적용하세요:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 \left(x + 1\right)} d x}}} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\left(\frac{\int{\frac{x^{\frac{5}{2}}}{x + 1} d x}}{6}\right)}}$$

$$$u=\sqrt{x}$$$라 하자.

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

적분은 다음과 같이 됩니다.

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{x^{\frac{5}{2}}}{x + 1} d x}}}}{6} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{2 u^{6}}{u^{2} + 1} d u}}}}{6}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=2$$$와 $$$f{\left(u \right)} = \frac{u^{6}}{u^{2} + 1}$$$에 적용하세요:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{2 u^{6}}{u^{2} + 1} d u}}}}{6} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\left(2 \int{\frac{u^{6}}{u^{2} + 1} d u}\right)}}}{6}$$

분자의 차수가 분모의 차수보다 크거나 같으므로 다항식의 긴 나눗셈을 수행하십시오(단계는 »에서 볼 수 있습니다):

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{u^{6}}{u^{2} + 1} d u}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\left(u^{4} - u^{2} + 1 - \frac{1}{u^{2} + 1}\right)d u}}}}{3}$$

각 항별로 적분하십시오:

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

상수 법칙 $$$\int c\, du = c u$$$을 $$$c=1$$$에 적용하십시오:

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

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

$$- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\int{u^{4} d u}}}}{3}=- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\frac{u^{1 + 4}}{1 + 4}}}}{3}=- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\left(\frac{u^{5}}{5}\right)}}}{3}$$

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

$$- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\int{u^{2} d u}}}}{3}=- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\frac{u^{1 + 2}}{1 + 2}}}}{3}=- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\left(\frac{u^{3}}{3}\right)}}}{3}$$

$$$\frac{1}{u^{2} + 1}$$$의 적분은 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

$$- \frac{u^{5}}{15} + \frac{u^{3}}{9} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{{\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{3} = - \frac{u^{5}}{15} + \frac{u^{3}}{9} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{{\color{red}{\operatorname{atan}{\left(u \right)}}}}{3}$$

다음 $$$u=\sqrt{x}$$$을 기억하라:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left({\color{red}{u}} \right)}}{3} - \frac{{\color{red}{u}}}{3} + \frac{{\color{red}{u}}^{3}}{9} - \frac{{\color{red}{u}}^{5}}{15} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left({\color{red}{\sqrt{x}}} \right)}}{3} - \frac{{\color{red}{\sqrt{x}}}}{3} + \frac{{\color{red}{\sqrt{x}}}^{3}}{9} - \frac{{\color{red}{\sqrt{x}}}^{5}}{15}$$

따라서,

$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}$$

적분 상수를 추가하세요:

$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}+C$$

$$$\int x^{2} \operatorname{atan}{\left(\sqrt{x} \right)}\, dx = \left(- \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}\right) + C$$$A