Integral of (2x+7)^8 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 x + 7$.

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

      $$\int \frac{u^{8}}{2}\, du$$

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

         $$\int u^{8}\, du = \frac{\int u^{8}\, du}{2}$$

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

            $$\int u^{8}\, du = \frac{u^{9}}{9}$$

         So, the result is: $\frac{u^{9}}{18}$

      Now substitute $u$ back in:

      $$\frac{\left(2 x + 7\right)^{9}}{18}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(2 x + 7\right)^{8} = 256 x^{8} + 7168 x^{7} + 87808 x^{6} + 614656 x^{5} + 2689120 x^{4} + 7529536 x^{3} + 13176688 x^{2} + 13176688 x + 5764801$$

   2. Integrate term-by-term:

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

         $$\int 256 x^{8}\, dx = 256 \int x^{8}\, dx$$

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

            $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

         So, the result is: $\frac{256 x^{9}}{9}$

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

         $$\int 7168 x^{7}\, dx = 7168 \int x^{7}\, dx$$

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

            $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

         So, the result is: $896 x^{8}$

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

         $$\int 87808 x^{6}\, dx = 87808 \int x^{6}\, dx$$

         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: $12544 x^{7}$

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

         $$\int 614656 x^{5}\, dx = 614656 \int x^{5}\, dx$$

         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: $\frac{307328 x^{6}}{3}$

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

         $$\int 2689120 x^{4}\, dx = 2689120 \int x^{4}\, dx$$

         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: $537824 x^{5}$

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

         $$\int 7529536 x^{3}\, dx = 7529536 \int x^{3}\, dx$$

         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: $1882384 x^{4}$

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

         $$\int 13176688 x^{2}\, dx = 13176688 \int x^{2}\, dx$$

         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: $\frac{13176688 x^{3}}{3}$

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

         $$\int 13176688 x\, dx = 13176688 \int x\, dx$$

         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: $6588344 x^{2}$

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

         $$\int 5764801\, dx = 5764801 x$$

      The result is: $\frac{256 x^{9}}{9} + 896 x^{8} + 12544 x^{7} + \frac{307328 x^{6}}{3} + 537824 x^{5} + 1882384 x^{4} + \frac{13176688 x^{3}}{3} + 6588344 x^{2} + 5764801 x$

2. Now simplify:

   $$\frac{\left(2 x + 7\right)^{9}}{18}$$

3. Add the constant of integration:

   $$\frac{\left(2 x + 7\right)^{9}}{18}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(2 x + 7\right)^{9}}{18}+ \mathrm{constant}$$