Integral of $$$\sin{\left(4 y_{} \right)}$$$

Find $$$\int \sin{\left(4 y_{} \right)}\, dy_{}$$$.

Let $$$u=4 y_{}$$$.

Then $$$du=\left(4 y_{}\right)^{\prime }dy_{} = 4 dy_{}$$$ (steps can be seen »), and we have that $$$dy_{} = \frac{du}{4}$$$.

The integral can be rewritten as

$${\color{red}{\int{\sin{\left(4 y_{} \right)} d y_{}}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{4} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}$$

Recall that $$$u=4 y_{}$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{4} = - \frac{\cos{\left({\color{red}{\left(4 y_{}\right)}} \right)}}{4}$$

Therefore,

$$\int{\sin{\left(4 y_{} \right)} d y_{}} = - \frac{\cos{\left(4 y_{} \right)}}{4}$$

Add the constant of integration:

$$\int{\sin{\left(4 y_{} \right)} d y_{}} = - \frac{\cos{\left(4 y_{} \right)}}{4}+C$$

$$$\int \sin{\left(4 y_{} \right)}\, dy_{} = - \frac{\cos{\left(4 y_{} \right)}}{4} + C$$$A