Integral of (5x-4)^4 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 5 x - 4$.

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{u^{4}}{25}\, du$$

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

         $$\int \frac{u^{4}}{5}\, du = \frac{\int u^{4}\, du}{5}$$

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

            $$\int u^{4}\, du = \frac{u^{5}}{5}$$

         So, the result is: $\frac{u^{5}}{25}$

      Now substitute $u$ back in:

      $$\frac{\left(5 x - 4\right)^{5}}{25}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 x - 4\right)^{4} = 625 x^{4} - 2000 x^{3} + 2400 x^{2} - 1280 x + 256$$

   2. Integrate term-by-term:

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

         $$\int 625 x^{4}\, dx = 625 \int x^{4}\, dx$$

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

            $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

         So, the result is: $125 x^{5}$

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

         $$\int \left(- 2000 x^{3}\right)\, dx = - 2000 \int x^{3}\, dx$$

         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: $- 500 x^{4}$

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

         $$\int 2400 x^{2}\, dx = 2400 \int x^{2}\, dx$$

         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: $800 x^{3}$

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

         $$\int \left(- 1280 x\right)\, dx = - 1280 \int x\, dx$$

         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: $- 640 x^{2}$

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

         $$\int 256\, dx = 256 x$$

      The result is: $125 x^{5} - 500 x^{4} + 800 x^{3} - 640 x^{2} + 256 x$

2. Now simplify:

   $$\frac{\left(5 x - 4\right)^{5}}{25}$$

3. Add the constant of integration:

   $$\frac{\left(5 x - 4\right)^{5}}{25}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(5 x - 4\right)^{5}}{25}+ \mathrm{constant}$$