Integral of $$$\frac{a - x}{\sqrt{x}}$$$ with respect to $$$x$$$

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

Expand the expression:

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

Integrate term by term:

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

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

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

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

$$- \frac{2 x^{\frac{3}{2}}}{3} + {\color{red}{\int{\frac{a}{\sqrt{x}} d x}}} = - \frac{2 x^{\frac{3}{2}}}{3} + {\color{red}{a \int{\frac{1}{\sqrt{x}} d x}}}$$

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

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

Therefore,

$$\int{\frac{a - x}{\sqrt{x}} d x} = 2 a \sqrt{x} - \frac{2 x^{\frac{3}{2}}}{3}$$

Simplify:

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

Add the constant of integration:

$$\int{\frac{a - x}{\sqrt{x}} d x} = \frac{2 \sqrt{x} \left(3 a - x\right)}{3}+C$$

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