Integral of $$$\sin{\left(\frac{\theta}{2} \right)}$$$

Find $$$\int \sin{\left(\frac{\theta}{2} \right)}\, d\theta$$$.

Let $$$u=\frac{\theta}{2}$$$.

Then $$$du=\left(\frac{\theta}{2}\right)^{\prime }d\theta = \frac{d\theta}{2}$$$ (steps can be seen »), and we have that $$$d\theta = 2 du$$$.

Therefore,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=2$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{2 \sin{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\sin{\left(u \right)} d u}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Recall that $$$u=\frac{\theta}{2}$$$:

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

Therefore,

$$\int{\sin{\left(\frac{\theta}{2} \right)} d \theta} = - 2 \cos{\left(\frac{\theta}{2} \right)}$$

Add the constant of integration:

$$\int{\sin{\left(\frac{\theta}{2} \right)} d \theta} = - 2 \cos{\left(\frac{\theta}{2} \right)}+C$$

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