Integral of 3x²+2x³ dx

1. Integrate term-by-term:

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

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

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

         $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

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

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

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

      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: $x^{3}$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is: