Integral of $$$y e^{x}$$$ with respect to $$$y$$$

Find $$$\int y e^{x}\, dy$$$.

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=e^{x}$$$ and $$$f{\left(y \right)} = y$$$:

$${\color{red}{\int{y e^{x} d y}}} = {\color{red}{e^{x} \int{y d y}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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