Integral of $$$\frac{\cos{\left(8 x \right)}}{2}$$$

Find $$$\int \frac{\cos{\left(8 x \right)}}{2}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \cos{\left(8 x \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(8 x \right)}}{2} d x}}} = {\color{red}{\left(\frac{\int{\cos{\left(8 x \right)} d x}}{2}\right)}}$$

Let $$$u=8 x$$$.

Then $$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{8}$$$.

The integral can be rewritten as

$$\frac{{\color{red}{\int{\cos{\left(8 x \right)} d x}}}}{2} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{2}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{8}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{2} = \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}}{2}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{16} = \frac{{\color{red}{\sin{\left(u \right)}}}}{16}$$

Recall that $$$u=8 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{16} = \frac{\sin{\left({\color{red}{\left(8 x\right)}} \right)}}{16}$$

Therefore,

$$\int{\frac{\cos{\left(8 x \right)}}{2} d x} = \frac{\sin{\left(8 x \right)}}{16}$$

Add the constant of integration:

$$\int{\frac{\cos{\left(8 x \right)}}{2} d x} = \frac{\sin{\left(8 x \right)}}{16}+C$$

$$$\int \frac{\cos{\left(8 x \right)}}{2}\, dx = \frac{\sin{\left(8 x \right)}}{16} + C$$$A