Integral of (2-x)^3 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 - x$.

      Then let $du = - dx$ and substitute $- du$:

      $$\int u^{3}\, du$$

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

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

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

            $$\int u^{3}\, du = \frac{u^{4}}{4}$$

         So, the result is: $- \frac{u^{4}}{4}$

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

      $$\left(2 - x\right)^{3} = - x^{3} + 6 x^{2} - 12 x + 8$$

   2. Integrate term-by-term:

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

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

         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: $- \frac{x^{4}}{4}$

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

         $$\int 6 x^{2}\, dx = 6 \int x^{2}\, dx$$

         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: $2 x^{3}$

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

         $$\int \left(- 12 x\right)\, dx = - 12 \int x\, dx$$

         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: $- 6 x^{2}$

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

         $$\int 8\, dx = 8 x$$

      The result is: $- \frac{x^{4}}{4} + 2 x^{3} - 6 x^{2} + 8 x$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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