Integral of $$$\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}$$$

Find $$$\int \left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)\, dy$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)d y}}} = {\color{red}{\left(\int{\frac{1}{2 \sqrt{y}} d y} + \int{\frac{\sqrt{y}}{2} d y}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(y \right)} = \sqrt{y}$$$:

$$\int{\frac{1}{2 \sqrt{y}} d y} + {\color{red}{\int{\frac{\sqrt{y}}{2} d y}}} = \int{\frac{1}{2 \sqrt{y}} d y} + {\color{red}{\left(\frac{\int{\sqrt{y} d y}}{2}\right)}}$$

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

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

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(y \right)} = \frac{1}{\sqrt{y}}$$$:

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

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

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

Therefore,

$$\int{\left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)d y} = \frac{y^{\frac{3}{2}}}{3} + \sqrt{y}$$

Simplify:

$$\int{\left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)d y} = \frac{\sqrt{y} \left(y + 3\right)}{3}$$

Add the constant of integration:

$$\int{\left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)d y} = \frac{\sqrt{y} \left(y + 3\right)}{3}+C$$

$$$\int \left(\frac{\sqrt{y}}{2} + \frac{1}{2 \sqrt{y}}\right)\, dy = \frac{\sqrt{y} \left(y + 3\right)}{3} + C$$$A