Integral of (1-2*x)^3 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 1 - 2 x$.

      Then let $du = - 2 dx$ and substitute $- \frac{du}{2}$:

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

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

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

         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: $- \frac{u^{4}}{8}$

      Now substitute $u$ back in:

      $$- \frac{\left(1 - 2 x\right)^{4}}{8}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(1 - 2 x\right)^{3} = - 8 x^{3} + 12 x^{2} - 6 x + 1$$

   2. Integrate term-by-term:

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

         $$\int \left(- 8 x^{3}\right)\, dx = - 8 \int x^{3}\, dx$$

         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: $- 2 x^{4}$

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

         $$\int 12 x^{2}\, dx = 12 \int x^{2}\, dx$$

         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: $4 x^{3}$

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

         $$\int \left(- 6 x\right)\, dx = - 6 \int x\, dx$$

         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: $- 2 x^{4} + 4 x^{3} - 3 x^{2} + x$

2. Now simplify:

   $$- \frac{\left(2 x - 1\right)^{4}}{8}$$

3. Add the constant of integration:

   $$- \frac{\left(2 x - 1\right)^{4}}{8}+ \mathrm{constant}$$

The answer is:

$$- \frac{\left(2 x - 1\right)^{4}}{8}+ \mathrm{constant}$$