Integral of $$$\sqrt{5 - \frac{x}{5}}$$$

Find $$$\int \sqrt{5 - \frac{x}{5}}\, dx$$$.

Let $$$u=5 - \frac{x}{5}$$$.

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

The integral becomes

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

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

$${\color{red}{\int{\left(- 5 \sqrt{u}\right)d u}}} = {\color{red}{\left(- 5 \int{\sqrt{u} d u}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{1}{2}$$$:

$$- 5 {\color{red}{\int{\sqrt{u} d u}}}=- 5 {\color{red}{\int{u^{\frac{1}{2}} d u}}}=- 5 {\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}=- 5 {\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}$$

Recall that $$$u=5 - \frac{x}{5}$$$:

$$- \frac{10 {\color{red}{u}}^{\frac{3}{2}}}{3} = - \frac{10 {\color{red}{\left(5 - \frac{x}{5}\right)}}^{\frac{3}{2}}}{3}$$

Therefore,

$$\int{\sqrt{5 - \frac{x}{5}} d x} = - \frac{10 \left(5 - \frac{x}{5}\right)^{\frac{3}{2}}}{3}$$

Simplify:

$$\int{\sqrt{5 - \frac{x}{5}} d x} = - \frac{2 \sqrt{5} \left(25 - x\right)^{\frac{3}{2}}}{15}$$

Add the constant of integration:

$$\int{\sqrt{5 - \frac{x}{5}} d x} = - \frac{2 \sqrt{5} \left(25 - x\right)^{\frac{3}{2}}}{15}+C$$

$$$\int \sqrt{5 - \frac{x}{5}}\, dx = - \frac{2 \sqrt{5} \left(25 - x\right)^{\frac{3}{2}}}{15} + C$$$A