Integral of (2sinx+6-3x^2) dx

1. Integrate term-by-term:

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

      $$\int \left(- 3 x^{2}\right)\, dx = - \int 3 x^{2}\, dx$$

      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: $x^{3}$

      So, the result is: $- x^{3}$

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

      $$\int 2 \sin{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)}\, dx$$

      1. The integral of sine is negative cosine:

         $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

      So, the result is: $- 2 \cos{\left(x \right)}$

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

      $$\int 6\, dx = 6 x$$

   The result is: $- x^{3} + 6 x - 2 \cos{\left(x \right)}$

2. Add the constant of integration:

   $$- x^{3} + 6 x - 2 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- x^{3} + 6 x - 2 \cos{\left(x \right)}+ \mathrm{constant}$$