Integral of (3*x-9)^6 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 3 x - 9$.

      Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{u^{6}}{3}\, du$$

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

         $$\int u^{6}\, du = \frac{\int u^{6}\, du}{3}$$

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

            $$\int u^{6}\, du = \frac{u^{7}}{7}$$

         So, the result is: $\frac{u^{7}}{21}$

      Now substitute $u$ back in:

      $$\frac{\left(3 x - 9\right)^{7}}{21}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(3 x - 9\right)^{6} = 729 x^{6} - 13122 x^{5} + 98415 x^{4} - 393660 x^{3} + 885735 x^{2} - 1062882 x + 531441$$

   2. Integrate term-by-term:

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

         $$\int 729 x^{6}\, dx = 729 \int x^{6}\, dx$$

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

            $$\int x^{6}\, dx = \frac{x^{7}}{7}$$

         So, the result is: $\frac{729 x^{7}}{7}$

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

         $$\int \left(- 13122 x^{5}\right)\, dx = - 13122 \int x^{5}\, dx$$

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

            $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

         So, the result is: $- 2187 x^{6}$

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

         $$\int 98415 x^{4}\, dx = 98415 \int x^{4}\, dx$$

         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: $19683 x^{5}$

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

         $$\int \left(- 393660 x^{3}\right)\, dx = - 393660 \int x^{3}\, dx$$

         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: $- 98415 x^{4}$

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

         $$\int 885735 x^{2}\, dx = 885735 \int x^{2}\, dx$$

         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: $295245 x^{3}$

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

         $$\int \left(- 1062882 x\right)\, dx = - 1062882 \int x\, dx$$

         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: $- 531441 x^{2}$

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

         $$\int 531441\, dx = 531441 x$$

      The result is: $\frac{729 x^{7}}{7} - 2187 x^{6} + 19683 x^{5} - 98415 x^{4} + 295245 x^{3} - 531441 x^{2} + 531441 x$

2. Now simplify:

   $$\frac{729 \left(x - 3\right)^{7}}{7}$$

3. Add the constant of integration:

   $$\frac{729 \left(x - 3\right)^{7}}{7}+ \mathrm{constant}$$

The answer is:

$$\frac{729 \left(x - 3\right)^{7}}{7}+ \mathrm{constant}$$