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

Find $$$\int \cos{\left(\frac{x}{2} \right)}\, dx$$$.

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

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

The integral becomes

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

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

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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