$$$144 \sin^{2}{\left(\theta \right)}$$$の積分

$$$\int 144 \sin^{2}{\left(\theta \right)}\, d\theta$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ を、$$$c=144$$$ と $$$f{\left(\theta \right)} = \sin^{2}{\left(\theta \right)}$$$ に対して適用する:

$${\color{red}{\int{144 \sin^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\left(144 \int{\sin^{2}{\left(\theta \right)} d \theta}\right)}}$$

冪低減公式 $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$ を $$$\alpha=\theta$$$ に適用する:

$$144 {\color{red}{\int{\sin^{2}{\left(\theta \right)} d \theta}}} = 144 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 \theta \right)}}{2}\right)d \theta}}}$$

定数倍の法則 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(\theta \right)} = 1 - \cos{\left(2 \theta \right)}$$$ に対して適用する:

$$144 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 \theta \right)}}{2}\right)d \theta}}} = 144 {\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 \theta \right)}\right)d \theta}}{2}\right)}}$$

項別に積分せよ:

$$72 {\color{red}{\int{\left(1 - \cos{\left(2 \theta \right)}\right)d \theta}}} = 72 {\color{red}{\left(\int{1 d \theta} - \int{\cos{\left(2 \theta \right)} d \theta}\right)}}$$

$$$c=1$$$ に対して定数則 $$$\int c\, d\theta = c \theta$$$ を適用する:

$$- 72 \int{\cos{\left(2 \theta \right)} d \theta} + 72 {\color{red}{\int{1 d \theta}}} = - 72 \int{\cos{\left(2 \theta \right)} d \theta} + 72 {\color{red}{\theta}}$$

$$$u=2 \theta$$$ とする。

すると $$$du=\left(2 \theta\right)^{\prime }d\theta = 2 d\theta$$$（手順は»で確認できます）、$$$d\theta = \frac{du}{2}$$$ となります。

この積分は次のように書き換えられる

$$72 \theta - 72 {\color{red}{\int{\cos{\left(2 \theta \right)} d \theta}}} = 72 \theta - 72 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=\frac{1}{2}$$$ と $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ に対して適用する:

$$72 \theta - 72 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 72 \theta - 72 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

余弦の積分は$$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$72 \theta - 36 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 72 \theta - 36 {\color{red}{\sin{\left(u \right)}}}$$

次のことを思い出してください $$$u=2 \theta$$$:

$$72 \theta - 36 \sin{\left({\color{red}{u}} \right)} = 72 \theta - 36 \sin{\left({\color{red}{\left(2 \theta\right)}} \right)}$$

したがって、

$$\int{144 \sin^{2}{\left(\theta \right)} d \theta} = 72 \theta - 36 \sin{\left(2 \theta \right)}$$

積分定数を加える:

$$\int{144 \sin^{2}{\left(\theta \right)} d \theta} = 72 \theta - 36 \sin{\left(2 \theta \right)}+C$$

$$$\int 144 \sin^{2}{\left(\theta \right)}\, d\theta = \left(72 \theta - 36 \sin{\left(2 \theta \right)}\right) + C$$$A