Integral of $$$x^{2} e^{4}$$$

Find $$$\int x^{2} e^{4}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{4}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{x^{2} e^{4} d x}}} = {\color{red}{e^{4} \int{x^{2} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$e^{4} {\color{red}{\int{x^{2} d x}}}=e^{4} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=e^{4} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{x^{2} e^{4} d x} = \frac{x^{3} e^{4}}{3}$$

Add the constant of integration:

$$\int{x^{2} e^{4} d x} = \frac{x^{3} e^{4}}{3}+C$$

$$$\int x^{2} e^{4}\, dx = \frac{x^{3} e^{4}}{3} + C$$$A