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

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

Let $$$u=2 \theta$$$.

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

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

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

Recall that $$$u=2 \theta$$$:

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

Therefore,

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

Add the constant of integration:

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

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