Integral of $$$y^{3} e^{x}$$$ with respect to $$$x$$$

Find $$$\int y^{3} e^{x}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=y^{3}$$$ and $$$f{\left(x \right)} = e^{x}$$$:

$${\color{red}{\int{y^{3} e^{x} d x}}} = {\color{red}{y^{3} \int{e^{x} d x}}}$$

The integral of the exponential function is $$$\int{e^{x} d x} = e^{x}$$$:

$$y^{3} {\color{red}{\int{e^{x} d x}}} = y^{3} {\color{red}{e^{x}}}$$

Therefore,

$$\int{y^{3} e^{x} d x} = y^{3} e^{x}$$

Add the constant of integration:

$$\int{y^{3} e^{x} d x} = y^{3} e^{x}+C$$

$$$\int y^{3} e^{x}\, dx = y^{3} e^{x} + C$$$A