Integral of $$$\frac{10 x^{2}}{e^{10}}$$$

Find $$$\int \frac{10 x^{2}}{e^{10}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{10}{e^{10}}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\frac{10 x^{2}}{e^{10}} d x}}} = {\color{red}{\left(\frac{10 \int{x^{2} d x}}{e^{10}}\right)}}$$

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

$$\frac{10 {\color{red}{\int{x^{2} d x}}}}{e^{10}}=\frac{10 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{e^{10}}=\frac{10 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{e^{10}}$$

Therefore,

$$\int{\frac{10 x^{2}}{e^{10}} d x} = \frac{10 x^{3}}{3 e^{10}}$$

Add the constant of integration:

$$\int{\frac{10 x^{2}}{e^{10}} d x} = \frac{10 x^{3}}{3 e^{10}}+C$$

$$$\int \frac{10 x^{2}}{e^{10}}\, dx = \frac{10 x^{3}}{3 e^{10}} + C$$$A