Integral of $$$\frac{\sqrt{u}}{5}$$$

Find $$$\int \frac{\sqrt{u}}{5}\, du$$$.

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

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

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

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

Therefore,

$$\int{\frac{\sqrt{u}}{5} d u} = \frac{2 u^{\frac{3}{2}}}{15}$$

Add the constant of integration:

$$\int{\frac{\sqrt{u}}{5} d u} = \frac{2 u^{\frac{3}{2}}}{15}+C$$

$$$\int \frac{\sqrt{u}}{5}\, du = \frac{2 u^{\frac{3}{2}}}{15} + C$$$A