Integral of (4x³+2x-3) dx

1. Integrate term-by-term:

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

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

      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: $x^{4}$

   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: $x^{2}$

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

      $$\int \left(\left(-1\right) 3\right)\, dx = - 3 x$$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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