Integral of $$$3 \sin{\left(5 x \right)}$$$

Find $$$\int 3 \sin{\left(5 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$:

$${\color{red}{\int{3 \sin{\left(5 x \right)} d x}}} = {\color{red}{\left(3 \int{\sin{\left(5 x \right)} d x}\right)}}$$

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

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

Therefore,

$$3 {\color{red}{\int{\sin{\left(5 x \right)} d x}}} = 3 {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}$$

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

$$3 {\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}} = 3 {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\sin{\left(u \right)} d u}}}}{5} = \frac{3 {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{5}$$

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

$$- \frac{3 \cos{\left({\color{red}{u}} \right)}}{5} = - \frac{3 \cos{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Therefore,

$$\int{3 \sin{\left(5 x \right)} d x} = - \frac{3 \cos{\left(5 x \right)}}{5}$$

Add the constant of integration:

$$\int{3 \sin{\left(5 x \right)} d x} = - \frac{3 \cos{\left(5 x \right)}}{5}+C$$

$$$\int 3 \sin{\left(5 x \right)}\, dx = - \frac{3 \cos{\left(5 x \right)}}{5} + C$$$A