Integral of (x-3)^4 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x - 3$.

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

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

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

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

      Now substitute $u$ back in:

      $$\frac{\left(x - 3\right)^{5}}{5}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(x - 3\right)^{4} = x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81$$

   2. Integrate term-by-term:

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

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

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

         $$\int \left(- 12 x^{3}\right)\, dx = - 12 \int x^{3}\, dx$$

         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: $- 3 x^{4}$

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

         $$\int 54 x^{2}\, dx = 54 \int x^{2}\, dx$$

         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: $18 x^{3}$

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

         $$\int \left(- 108 x\right)\, dx = - 108 \int x\, dx$$

         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: $- 54 x^{2}$

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

         $$\int 81\, dx = 81 x$$

      The result is: $\frac{x^{5}}{5} - 3 x^{4} + 18 x^{3} - 54 x^{2} + 81 x$

2. Now simplify:

   $$\frac{\left(x - 3\right)^{5}}{5}$$

3. Add the constant of integration:

   $$\frac{\left(x - 3\right)^{5}}{5}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x - 3\right)^{5}}{5}+ \mathrm{constant}$$