Integral of $$$x^{4} e^{6}$$$

Find $$$\int x^{4} e^{6}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{6}$$$ and $$$f{\left(x \right)} = x^{4}$$$:

$${\color{red}{\int{x^{4} e^{6} d x}}} = {\color{red}{e^{6} \int{x^{4} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$e^{6} {\color{red}{\int{x^{4} d x}}}=e^{6} {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=e^{6} {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Therefore,

$$\int{x^{4} e^{6} d x} = \frac{x^{5} e^{6}}{5}$$

Add the constant of integration:

$$\int{x^{4} e^{6} d x} = \frac{x^{5} e^{6}}{5}+C$$

$$$\int x^{4} e^{6}\, dx = \frac{x^{5} e^{6}}{5} + C$$$A