Integral of $$$4 x^{3} e^{4}$$$

Find $$$\int 4 x^{3} e^{4}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4 e^{4}$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$${\color{red}{\int{4 x^{3} e^{4} d x}}} = {\color{red}{\left(4 e^{4} \int{x^{3} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$4 e^{4} {\color{red}{\int{x^{3} d x}}}=4 e^{4} {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=4 e^{4} {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Therefore,

$$\int{4 x^{3} e^{4} d x} = x^{4} e^{4}$$

Add the constant of integration:

$$\int{4 x^{3} e^{4} d x} = x^{4} e^{4}+C$$

$$$\int 4 x^{3} e^{4}\, dx = x^{4} e^{4} + C$$$A