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

Find $$$\int \frac{\sin{\left(5 x \right)}}{5}\, dx$$$.

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

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

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

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

Thus,

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

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 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

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

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{25} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{25}$$

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

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{25} = - \frac{\cos{\left({\color{red}{\left(5 x\right)}} \right)}}{25}$$

Therefore,

$$\int{\frac{\sin{\left(5 x \right)}}{5} d x} = - \frac{\cos{\left(5 x \right)}}{25}$$

Add the constant of integration:

$$\int{\frac{\sin{\left(5 x \right)}}{5} d x} = - \frac{\cos{\left(5 x \right)}}{25}+C$$

$$$\int \frac{\sin{\left(5 x \right)}}{5}\, dx = - \frac{\cos{\left(5 x \right)}}{25} + C$$$A