Integral of $$$x^{5} e^{3}$$$

Find $$$\int x^{5} e^{3}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{3}$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$${\color{red}{\int{x^{5} e^{3} d x}}} = {\color{red}{e^{3} \int{x^{5} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$e^{3} {\color{red}{\int{x^{5} d x}}}=e^{3} {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=e^{3} {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Therefore,

$$\int{x^{5} e^{3} d x} = \frac{x^{6} e^{3}}{6}$$

Add the constant of integration:

$$\int{x^{5} e^{3} d x} = \frac{x^{6} e^{3}}{6}+C$$

$$$\int x^{5} e^{3}\, dx = \frac{x^{6} e^{3}}{6} + C$$$A