Integral of ((6-3x)^2)/2 dx

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

   $$\int \frac{\left(6 - 3 x\right)^{2}}{2}\, dx = \frac{\int \left(6 - 3 x\right)^{2}\, dx}{2}$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = 6 - 3 x$.

         Then let $du = - 3 dx$ and substitute $- \frac{du}{3}$:

         $$\int \left(- \frac{u^{2}}{3}\right)\, du$$

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

            $$\int u^{2}\, du = - \frac{\int u^{2}\, du}{3}$$

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{9}$

         Now substitute $u$ back in:

         $$- \frac{\left(6 - 3 x\right)^{3}}{9}$$

      Method #2

      1. Rewrite the integrand:

         $$\left(6 - 3 x\right)^{2} = 9 x^{2} - 36 x + 36$$

      2. Integrate term-by-term:

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

            $$\int 9 x^{2}\, dx = 9 \int x^{2}\, dx$$

            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: $3 x^{3}$

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

            $$\int \left(- 36 x\right)\, dx = - 36 \int x\, dx$$

            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: $- 18 x^{2}$

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

            $$\int 36\, dx = 36 x$$

         The result is: $3 x^{3} - 18 x^{2} + 36 x$

   So, the result is: $- \frac{\left(6 - 3 x\right)^{3}}{18}$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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