Integral of $$$\cos{\left(5 x \right)}$$$

Let $$$u=5 x$$$.

Then $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{5}$$$.

So,

$${\color{red}{\int{\cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{5} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{5} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{5}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{5} = \frac{{\color{red}{\sin{\left(u \right)}}}}{5}$$

Recall that $$$u=5 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{5} = \frac{\sin{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Therefore,

$$\int{\cos{\left(5 x \right)} d x} = \frac{\sin{\left(5 x \right)}}{5}$$

Add the constant of integration:

$$\int{\cos{\left(5 x \right)} d x} = \frac{\sin{\left(5 x \right)}}{5}+C$$

$$$\int \cos{\left(5 x \right)}\, dx = \frac{\sin{\left(5 x \right)}}{5} + C$$$A