Integral of (12x¾-9x^5/3)dx dx

1. Integrate term-by-term:

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

      $$\int \frac{3 \cdot 12 x}{4}\, dx = \frac{3 \int 12 x\, dx}{4}$$

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

         $$\int 12 x\, dx = 12 \int x\, dx$$

         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: $6 x^{2}$

      So, the result is: $\frac{9 x^{2}}{2}$

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

      $$\int \left(- \frac{9 x^{5}}{3}\right)\, dx = - \frac{\int 9 x^{5}\, dx}{3}$$

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

         $$\int 9 x^{5}\, dx = 9 \int x^{5}\, dx$$

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

            $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

         So, the result is: $\frac{3 x^{6}}{2}$

      So, the result is: $- \frac{x^{6}}{2}$

   The result is: $- \frac{x^{6}}{2} + \frac{9 x^{2}}{2}$

2. Now simplify:

   $$\frac{x^{2} \left(9 - x^{4}\right)}{2}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(9 - x^{4}\right)}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(9 - x^{4}\right)}{2}+ \mathrm{constant}$$