Integral of 3x^2+x-4 dx

1. Integrate term-by-term:

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

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

      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 $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

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

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

      $$\int \left(\left(-1\right) 4\right)\, dx = - 4 x$$

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

2. Now simplify:

   $$\frac{x \left(2 x^{2} + x - 8\right)}{2}$$

3. Add the constant of integration:

   $$\frac{x \left(2 x^{2} + x - 8\right)}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(2 x^{2} + x - 8\right)}{2}+ \mathrm{constant}$$