Integral of 3*x^2+2*x+5 dx

1. Integrate term-by-term:

   1. Integrate term-by-term:

      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: $x^{3}$

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

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

         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: $x^{2}$

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

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

      $$\int 5\, dx = 5 x$$

   The result is: $x^{3} + x^{2} + 5 x$

2. Now simplify:

   $$x \left(x^{2} + x + 5\right)$$

3. Add the constant of integration:

   $$x \left(x^{2} + x + 5\right)+ \mathrm{constant}$$

The answer is:

$$x \left(x^{2} + x + 5\right)+ \mathrm{constant}$$