Integral of xe-x^2 dx

1. Integrate term-by-term:

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

      $$\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx$$

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

         $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

      So, the result is: $- \frac{x^{3}}{3}$

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

      $$\int e x\, dx = e \int x\, dx$$

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

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

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

   The result is: $- \frac{x^{3}}{3} + \frac{e x^{2}}{2}$

2. Now simplify:

   $$x^{2} \left(- \frac{x}{3} + \frac{e}{2}\right)$$

3. Add the constant of integration:

   $$x^{2} \left(- \frac{x}{3} + \frac{e}{2}\right)+ \mathrm{constant}$$

The answer is:

$$x^{2} \left(- \frac{x}{3} + \frac{e}{2}\right)+ \mathrm{constant}$$