$$$16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)}$$$의 적분

$$$\int 16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)}\, d\theta$$$을(를) 구하시오.

멱 감소 공식 $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$를 $$$\alpha=\theta$$$에 적용하세요:

$${\color{red}{\int{16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)} d \theta}}} = {\color{red}{\int{2 \left(4 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)} + 3\right) \sin{\left(\theta \right)} d \theta}}}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=\frac{1}{8}$$$와 $$$f{\left(\theta \right)} = 16 \left(4 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)} + 3\right) \sin{\left(\theta \right)}$$$에 적용하세요:

$${\color{red}{\int{2 \left(4 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)} + 3\right) \sin{\left(\theta \right)} d \theta}}} = {\color{red}{\left(\frac{\int{16 \left(4 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)} + 3\right) \sin{\left(\theta \right)} d \theta}}{8}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{16 \left(4 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)} + 3\right) \sin{\left(\theta \right)} d \theta}}}}{8} = \frac{{\color{red}{\int{\left(64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} + 16 \sin{\left(\theta \right)} \cos{\left(4 \theta \right)} + 48 \sin{\left(\theta \right)}\right)d \theta}}}}{8}$$

각 항별로 적분하십시오:

$$\frac{{\color{red}{\int{\left(64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} + 16 \sin{\left(\theta \right)} \cos{\left(4 \theta \right)} + 48 \sin{\left(\theta \right)}\right)d \theta}}}}{8} = \frac{{\color{red}{\left(\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta} + \int{16 \sin{\left(\theta \right)} \cos{\left(4 \theta \right)} d \theta} + \int{48 \sin{\left(\theta \right)} d \theta}\right)}}}{8}$$

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=48$$$와 $$$f{\left(\theta \right)} = \sin{\left(\theta \right)}$$$에 적용하세요:

$$\frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + \frac{\int{16 \sin{\left(\theta \right)} \cos{\left(4 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\int{48 \sin{\left(\theta \right)} d \theta}}}}{8} = \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + \frac{\int{16 \sin{\left(\theta \right)} \cos{\left(4 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\left(48 \int{\sin{\left(\theta \right)} d \theta}\right)}}}{8}$$

사인 함수의 적분은 $$$\int{\sin{\left(\theta \right)} d \theta} = - \cos{\left(\theta \right)}$$$:

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

공식 $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$에 $$$\alpha=\theta$$$와 $$$\beta=4 \theta$$$를 대입하여 $$$\sin\left(\theta \right)\cos\left(4 \theta \right)$$$을(를) 다시 쓰십시오.:

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

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=\frac{1}{2}$$$와 $$$f{\left(\theta \right)} = - 16 \sin{\left(3 \theta \right)} + 16 \sin{\left(5 \theta \right)}$$$에 적용하세요:

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

각 항별로 적분하십시오:

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

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=16$$$와 $$$f{\left(\theta \right)} = \sin{\left(3 \theta \right)}$$$에 적용하세요:

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

$$$u=3 \theta$$$라 하자.

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

적분은 다음과 같이 됩니다.

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

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=\frac{1}{3}$$$와 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$에 적용하세요:

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

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

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

다음 $$$u=3 \theta$$$을 기억하라:

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

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=16$$$와 $$$f{\left(\theta \right)} = \sin{\left(5 \theta \right)}$$$에 적용하세요:

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

$$$u=5 \theta$$$라 하자.

그러면 $$$du=\left(5 \theta\right)^{\prime }d\theta = 5 d\theta$$$ (단계는 »에서 볼 수 있습니다), 그리고 $$$d\theta = \frac{du}{5}$$$임을 얻습니다.

적분은 다음과 같이 됩니다.

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + {\color{red}{\int{\sin{\left(5 \theta \right)} d \theta}}} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}$$

상수배 법칙 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$을 $$$c=\frac{1}{5}$$$와 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$에 적용하세요:

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}$$

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

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{5} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} + \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{5}$$

다음 $$$u=5 \theta$$$을 기억하라:

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} - \frac{\cos{\left({\color{red}{u}} \right)}}{5} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} + \frac{\int{64 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} d \theta}}{8} - \frac{\cos{\left({\color{red}{\left(5 \theta\right)}} \right)}}{5}$$

공식 $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$에 $$$\alpha=\theta$$$와 $$$\beta=2 \theta$$$를 대입하여 $$$\sin\left(\theta \right)\cos\left(2 \theta \right)$$$을(를) 다시 쓰십시오.:

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

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=\frac{1}{2}$$$와 $$$f{\left(\theta \right)} = - 64 \sin{\left(\theta \right)} + 64 \sin{\left(3 \theta \right)}$$$에 적용하세요:

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

각 항별로 적분하십시오:

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{{\color{red}{\int{\left(- 64 \sin{\left(\theta \right)} + 64 \sin{\left(3 \theta \right)}\right)d \theta}}}}{16} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{{\color{red}{\left(- \int{64 \sin{\left(\theta \right)} d \theta} + \int{64 \sin{\left(3 \theta \right)} d \theta}\right)}}}{16}$$

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

$$- 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{\int{64 \sin{\left(3 \theta \right)} d \theta}}{16} - \frac{{\color{red}{\int{64 \sin{\left(\theta \right)} d \theta}}}}{16} = - 6 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{\int{64 \sin{\left(3 \theta \right)} d \theta}}{16} - \frac{{\color{red}{\left(64 \int{\sin{\left(\theta \right)} d \theta}\right)}}}{16}$$

사인 함수의 적분은 $$$\int{\sin{\left(\theta \right)} d \theta} = - \cos{\left(\theta \right)}$$$:

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

상수배 법칙 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$을 $$$c=64$$$와 $$$f{\left(\theta \right)} = \sin{\left(3 \theta \right)}$$$에 적용하세요:

$$- 2 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{{\color{red}{\int{64 \sin{\left(3 \theta \right)} d \theta}}}}{16} = - 2 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + \frac{{\color{red}{\left(64 \int{\sin{\left(3 \theta \right)} d \theta}\right)}}}{16}$$

이미 계산된 적분 $$$\int{\sin{\left(3 \theta \right)} d \theta}$$$:

$$\int{\sin{\left(3 \theta \right)} d \theta} = - \frac{\cos{\left(3 \theta \right)}}{3}$$

따라서,

$$- 2 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + 4 {\color{red}{\int{\sin{\left(3 \theta \right)} d \theta}}} = - 2 \cos{\left(\theta \right)} + \frac{\cos{\left(3 \theta \right)}}{3} - \frac{\cos{\left(5 \theta \right)}}{5} + 4 {\color{red}{\left(- \frac{\cos{\left(3 \theta \right)}}{3}\right)}}$$

따라서,

$$\int{16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)} d \theta} = - 2 \cos{\left(\theta \right)} - \cos{\left(3 \theta \right)} - \frac{\cos{\left(5 \theta \right)}}{5}$$

적분 상수를 추가하세요:

$$\int{16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)} d \theta} = - 2 \cos{\left(\theta \right)} - \cos{\left(3 \theta \right)} - \frac{\cos{\left(5 \theta \right)}}{5}+C$$

$$$\int 16 \sin{\left(\theta \right)} \cos^{4}{\left(\theta \right)}\, d\theta = \left(- 2 \cos{\left(\theta \right)} - \cos{\left(3 \theta \right)} - \frac{\cos{\left(5 \theta \right)}}{5}\right) + C$$$A