Integral of 1+6*x-3*x^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:

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

      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. Integrate term-by-term:

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

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

         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: $3 x^{2}$

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

         $$\int 1\, dx = x$$

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

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

2. Now simplify:

   $$x \left(- x^{2} + 3 x + 1\right)$$

3. Add the constant of integration:

   $$x \left(- x^{2} + 3 x + 1\right)+ \mathrm{constant}$$

The answer is:

$$x \left(- x^{2} + 3 x + 1\right)+ \mathrm{constant}$$