Integral of x*e^3x^2+2 dx

1. Integrate term-by-term:

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

      Then let $du = 2 x dx$ and substitute $\frac{du e^{3}}{2}$:

      $$\int \frac{u e^{3}}{2}\, du$$

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

         $$\int u\, du = \frac{e^{3} \int u\, du}{2}$$

         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: $\frac{u^{2} e^{3}}{4}$

      Now substitute $u$ back in:

      $$\frac{x^{4} e^{3}}{4}$$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 2\, dx = 2 x$$

   The result is: $\frac{x^{4} e^{3}}{4} + 2 x$

2. Now simplify:

   $$\frac{x \left(x^{3} e^{3} + 8\right)}{4}$$

3. Add the constant of integration:

   $$\frac{x \left(x^{3} e^{3} + 8\right)}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(x^{3} e^{3} + 8\right)}{4}+ \mathrm{constant}$$