Integral of $$$\frac{x}{y}$$$ with respect to $$$x$$$

Find $$$\int \frac{x}{y}\, dx$$$.

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

$${\color{red}{\int{\frac{x}{y} d x}}} = {\color{red}{\frac{\int{x d x}}{y}}}$$

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

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

Therefore,

$$\int{\frac{x}{y} d x} = \frac{x^{2}}{2 y}$$

Add the constant of integration:

$$\int{\frac{x}{y} d x} = \frac{x^{2}}{2 y}+C$$

$$$\int \frac{x}{y}\, dx = \frac{x^{2}}{2 y} + C$$$A