Integral of $$$s \sin{\left(10 x \right)}$$$ with respect to $$$x$$$

Find $$$\int s \sin{\left(10 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=s$$$ and $$$f{\left(x \right)} = \sin{\left(10 x \right)}$$$:

$${\color{red}{\int{s \sin{\left(10 x \right)} d x}}} = {\color{red}{s \int{\sin{\left(10 x \right)} d x}}}$$

Let $$$u=10 x$$$.

Then $$$du=\left(10 x\right)^{\prime }dx = 10 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{10}$$$.

Thus,

$$s {\color{red}{\int{\sin{\left(10 x \right)} d x}}} = s {\color{red}{\int{\frac{\sin{\left(u \right)}}{10} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{10}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$s {\color{red}{\int{\frac{\sin{\left(u \right)}}{10} d u}}} = s {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{10}\right)}}$$

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

$$\frac{s {\color{red}{\int{\sin{\left(u \right)} d u}}}}{10} = \frac{s {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{10}$$

Recall that $$$u=10 x$$$:

$$- \frac{s \cos{\left({\color{red}{u}} \right)}}{10} = - \frac{s \cos{\left({\color{red}{\left(10 x\right)}} \right)}}{10}$$

Therefore,

$$\int{s \sin{\left(10 x \right)} d x} = - \frac{s \cos{\left(10 x \right)}}{10}$$

Add the constant of integration:

$$\int{s \sin{\left(10 x \right)} d x} = - \frac{s \cos{\left(10 x \right)}}{10}+C$$

$$$\int s \sin{\left(10 x \right)}\, dx = - \frac{s \cos{\left(10 x \right)}}{10} + C$$$A