Integral of (5-2x)³ dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 5 - 2 x$.

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

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

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

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

         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:

      $$- \frac{\left(5 - 2 x\right)^{4}}{8}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 - 2 x\right)^{3} = - 8 x^{3} + 60 x^{2} - 150 x + 125$$

   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:

         $$\int 60 x^{2}\, dx = 60 \int x^{2}\, dx$$

         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: $20 x^{3}$

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

         $$\int \left(- 150 x\right)\, dx = - 150 \int x\, dx$$

         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: $- 75 x^{2}$

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

         $$\int 125\, dx = 125 x$$

      The result is: $- 2 x^{4} + 20 x^{3} - 75 x^{2} + 125 x$

2. Now simplify:

   $$- \frac{\left(2 x - 5\right)^{4}}{8}$$

3. Add the constant of integration:

   $$- \frac{\left(2 x - 5\right)^{4}}{8}+ \mathrm{constant}$$

The answer is:

$$- \frac{\left(2 x - 5\right)^{4}}{8}+ \mathrm{constant}$$