Integral of $$$\frac{t y}{e}$$$ with respect to $$$y$$$

Find $$$\int \frac{t y}{e}\, dy$$$.

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

$${\color{red}{\int{\frac{t y}{e} d y}}} = {\color{red}{\frac{t \int{y d y}}{e}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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