Integral of 4(2x-1)^2 dx

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

   $$\int 4 \left(2 x - 1\right)^{2}\, dx = 4 \int \left(2 x - 1\right)^{2}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

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

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

         $$\int \frac{u^{2}}{4}\, du$$

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

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

            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: $\frac{u^{3}}{6}$

         Now substitute $u$ back in:

         $$\frac{\left(2 x - 1\right)^{3}}{6}$$

      Method #2

      1. Rewrite the integrand:

         $$\left(2 x - 1\right)^{2} = 4 x^{2} - 4 x + 1$$

      2. Integrate term-by-term:

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

            $$\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$$

            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: $\frac{4 x^{3}}{3}$

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

            $$\int \left(- 4 x\right)\, dx = - 4 \int x\, dx$$

            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: $- 2 x^{2}$

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

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

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

   So, the result is: $\frac{2 \left(2 x - 1\right)^{3}}{3}$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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