$$$\sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)}$$$의 적분

$$$\int \sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)}\, d\theta$$$을(를) 구하시오.

배각 공식 $$$\sin\left(\theta \right)\cos\left(\theta \right)=\frac{1}{2}\sin\left( 2 \theta \right)$$$을(를) 사용하여 피적분함수를 다시 쓰십시오:

$${\color{red}{\int{\sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)} d \theta}}} = {\color{red}{\int{\frac{\sin^{6}{\left(2 \theta \right)}}{64} d \theta}}}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=\frac{1}{64}$$$와 $$$f{\left(\theta \right)} = \sin^{6}{\left(2 \theta \right)}$$$에 적용하세요:

$${\color{red}{\int{\frac{\sin^{6}{\left(2 \theta \right)}}{64} d \theta}}} = {\color{red}{\left(\frac{\int{\sin^{6}{\left(2 \theta \right)} d \theta}}{64}\right)}}$$

멱 감소 공식 $$$\sin^{6}{\left(\alpha \right)} = - \frac{15 \cos{\left(2 \alpha \right)}}{32} + \frac{3 \cos{\left(4 \alpha \right)}}{16} - \frac{\cos{\left(6 \alpha \right)}}{32} + \frac{5}{16}$$$를 $$$\alpha=2 \theta$$$에 적용하세요:

$$\frac{{\color{red}{\int{\sin^{6}{\left(2 \theta \right)} d \theta}}}}{64} = \frac{{\color{red}{\int{\left(- \frac{15 \cos{\left(4 \theta \right)}}{32} + \frac{3 \cos{\left(8 \theta \right)}}{16} - \frac{\cos{\left(12 \theta \right)}}{32} + \frac{5}{16}\right)d \theta}}}}{64}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=\frac{1}{32}$$$와 $$$f{\left(\theta \right)} = - 15 \cos{\left(4 \theta \right)} + 6 \cos{\left(8 \theta \right)} - \cos{\left(12 \theta \right)} + 10$$$에 적용하세요:

$$\frac{{\color{red}{\int{\left(- \frac{15 \cos{\left(4 \theta \right)}}{32} + \frac{3 \cos{\left(8 \theta \right)}}{16} - \frac{\cos{\left(12 \theta \right)}}{32} + \frac{5}{16}\right)d \theta}}}}{64} = \frac{{\color{red}{\left(\frac{\int{\left(- 15 \cos{\left(4 \theta \right)} + 6 \cos{\left(8 \theta \right)} - \cos{\left(12 \theta \right)} + 10\right)d \theta}}{32}\right)}}}{64}$$

각 항별로 적분하십시오:

$$\frac{{\color{red}{\int{\left(- 15 \cos{\left(4 \theta \right)} + 6 \cos{\left(8 \theta \right)} - \cos{\left(12 \theta \right)} + 10\right)d \theta}}}}{2048} = \frac{{\color{red}{\left(\int{10 d \theta} - \int{15 \cos{\left(4 \theta \right)} d \theta} + \int{6 \cos{\left(8 \theta \right)} d \theta} - \int{\cos{\left(12 \theta \right)} d \theta}\right)}}}{2048}$$

상수 법칙 $$$\int c\, d\theta = c \theta$$$을 $$$c=10$$$에 적용하십시오:

$$- \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{\int{\cos{\left(12 \theta \right)} d \theta}}{2048} + \frac{{\color{red}{\int{10 d \theta}}}}{2048} = - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{\int{\cos{\left(12 \theta \right)} d \theta}}{2048} + \frac{{\color{red}{\left(10 \theta\right)}}}{2048}$$

$$$u=12 \theta$$$라 하자.

그러면 $$$du=\left(12 \theta\right)^{\prime }d\theta = 12 d\theta$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$d\theta = \frac{du}{12}$$$임을 얻습니다.

따라서,

$$\frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\int{\cos{\left(12 \theta \right)} d \theta}}}}{2048} = \frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{12} d u}}}}{2048}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=\frac{1}{12}$$$와 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$에 적용하세요:

$$\frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{12} d u}}}}{2048} = \frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{12}\right)}}}{2048}$$

코사인의 적분은 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{24576} = \frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\sin{\left(u \right)}}}}{24576}$$

다음 $$$u=12 \theta$$$을 기억하라:

$$\frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{\sin{\left({\color{red}{u}} \right)}}{24576} = \frac{5 \theta}{1024} - \frac{\int{15 \cos{\left(4 \theta \right)} d \theta}}{2048} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{\sin{\left({\color{red}{\left(12 \theta\right)}} \right)}}{24576}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=15$$$와 $$$f{\left(\theta \right)} = \cos{\left(4 \theta \right)}$$$에 적용하세요:

$$\frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\int{15 \cos{\left(4 \theta \right)} d \theta}}}}{2048} = \frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{{\color{red}{\left(15 \int{\cos{\left(4 \theta \right)} d \theta}\right)}}}{2048}$$

$$$u=4 \theta$$$라 하자.

그러면 $$$du=\left(4 \theta\right)^{\prime }d\theta = 4 d\theta$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$d\theta = \frac{du}{4}$$$임을 얻습니다.

적분은 다음과 같이 다시 쓸 수 있습니다.

$$\frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\int{\cos{\left(4 \theta \right)} d \theta}}}}{2048} = \frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2048}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=\frac{1}{4}$$$와 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$에 적용하세요:

$$\frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2048} = \frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2048}$$

코사인의 적분은 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{8192} = \frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 {\color{red}{\sin{\left(u \right)}}}}{8192}$$

다음 $$$u=4 \theta$$$을 기억하라:

$$\frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 \sin{\left({\color{red}{u}} \right)}}{8192} = \frac{5 \theta}{1024} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{\int{6 \cos{\left(8 \theta \right)} d \theta}}{2048} - \frac{15 \sin{\left({\color{red}{\left(4 \theta\right)}} \right)}}{8192}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=6$$$와 $$$f{\left(\theta \right)} = \cos{\left(8 \theta \right)}$$$에 적용하세요:

$$\frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{{\color{red}{\int{6 \cos{\left(8 \theta \right)} d \theta}}}}{2048} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{{\color{red}{\left(6 \int{\cos{\left(8 \theta \right)} d \theta}\right)}}}{2048}$$

$$$u=8 \theta$$$라 하자.

그러면 $$$du=\left(8 \theta\right)^{\prime }d\theta = 8 d\theta$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$d\theta = \frac{du}{8}$$$임을 얻습니다.

따라서,

$$\frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\int{\cos{\left(8 \theta \right)} d \theta}}}}{1024} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{1024}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=\frac{1}{8}$$$와 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$에 적용하세요:

$$\frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{1024} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}}{1024}$$

코사인의 적분은 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{8192} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 {\color{red}{\sin{\left(u \right)}}}}{8192}$$

다음 $$$u=8 \theta$$$을 기억하라:

$$\frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 \sin{\left({\color{red}{u}} \right)}}{8192} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576} + \frac{3 \sin{\left({\color{red}{\left(8 \theta\right)}} \right)}}{8192}$$

따라서,

$$\int{\sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)} d \theta} = \frac{5 \theta}{1024} - \frac{15 \sin{\left(4 \theta \right)}}{8192} + \frac{3 \sin{\left(8 \theta \right)}}{8192} - \frac{\sin{\left(12 \theta \right)}}{24576}$$

간단히 하시오:

$$\int{\sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)} d \theta} = - \frac{- 120 \theta + 45 \sin{\left(4 \theta \right)} - 9 \sin{\left(8 \theta \right)} + \sin{\left(12 \theta \right)}}{24576}$$

적분 상수를 추가하세요:

$$\int{\sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)} d \theta} = - \frac{- 120 \theta + 45 \sin{\left(4 \theta \right)} - 9 \sin{\left(8 \theta \right)} + \sin{\left(12 \theta \right)}}{24576}+C$$

$$$\int \sin^{6}{\left(\theta \right)} \cos^{6}{\left(\theta \right)}\, d\theta = - \frac{- 120 \theta + 45 \sin{\left(4 \theta \right)} - 9 \sin{\left(8 \theta \right)} + \sin{\left(12 \theta \right)}}{24576} + C$$$A