Integral of x^2e^-2xdx dx

1. The integral of a constant times a function is the constant times the integral of the function:

   $$\int x^{2} \frac{1}{e^{2}} x 1\, dx = \frac{\int x x^{2}\, dx}{e^{2}}$$

   1. Let $u = x^{2}$.

      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$:

      $$\int \frac{u}{4}\, du$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{u}{2}\, du = \frac{\int u\, du}{2}$$

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int u\, du = \frac{u^{2}}{2}$$

         So, the result is: $\frac{u^{2}}{4}$

      Now substitute $u$ back in:

      $$\frac{x^{4}}{4}$$

   So, the result is: $\frac{x^{4}}{4 e^{2}}$

2. Add the constant of integration:

   $$\frac{x^{4}}{4 e^{2}}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{4}}{4 e^{2}}+ \mathrm{constant}$$