Integral of $$$\frac{\sqrt{10} \sqrt{x}}{z}$$$ with respect to $$$x$$$

Find $$$\int \frac{\sqrt{10} \sqrt{x}}{z}\, dx$$$.

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

$${\color{red}{\int{\frac{\sqrt{10} \sqrt{x}}{z} d x}}} = {\color{red}{\frac{\sqrt{10} \int{\sqrt{x} d x}}{z}}}$$

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

$$\frac{\sqrt{10} {\color{red}{\int{\sqrt{x} d x}}}}{z}=\frac{\sqrt{10} {\color{red}{\int{x^{\frac{1}{2}} d x}}}}{z}=\frac{\sqrt{10} {\color{red}{\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{z}=\frac{\sqrt{10} {\color{red}{\left(\frac{2 x^{\frac{3}{2}}}{3}\right)}}}{z}$$

Therefore,

$$\int{\frac{\sqrt{10} \sqrt{x}}{z} d x} = \frac{2 \sqrt{10} x^{\frac{3}{2}}}{3 z}$$

Add the constant of integration:

$$\int{\frac{\sqrt{10} \sqrt{x}}{z} d x} = \frac{2 \sqrt{10} x^{\frac{3}{2}}}{3 z}+C$$

$$$\int \frac{\sqrt{10} \sqrt{x}}{z}\, dx = \frac{2 \sqrt{10} x^{\frac{3}{2}}}{3 z} + C$$$A