Integral of $$$i x^{2} e^{3}$$$

Find $$$\int i x^{2} e^{3}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=i e^{3}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{i x^{2} e^{3} d x}}} = {\color{red}{i e^{3} \int{x^{2} d x}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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