Integral of (2x+3)^7 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 x + 3$.

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

      $$\int \frac{u^{7}}{2}\, du$$

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

         $$\int u^{7}\, du = \frac{\int u^{7}\, du}{2}$$

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

            $$\int u^{7}\, du = \frac{u^{8}}{8}$$

         So, the result is: $\frac{u^{8}}{16}$

      Now substitute $u$ back in:

      $$\frac{\left(2 x + 3\right)^{8}}{16}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(2 x + 3\right)^{7} = 128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187$$

   2. Integrate term-by-term:

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

         $$\int 128 x^{7}\, dx = 128 \int x^{7}\, dx$$

         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: $16 x^{8}$

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

         $$\int 1344 x^{6}\, dx = 1344 \int x^{6}\, dx$$

         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: $192 x^{7}$

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

         $$\int 6048 x^{5}\, dx = 6048 \int x^{5}\, dx$$

         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: $1008 x^{6}$

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

         $$\int 15120 x^{4}\, dx = 15120 \int x^{4}\, dx$$

         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: $3024 x^{5}$

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

         $$\int 22680 x^{3}\, dx = 22680 \int x^{3}\, dx$$

         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: $5670 x^{4}$

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

         $$\int 20412 x^{2}\, dx = 20412 \int x^{2}\, dx$$

         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: $6804 x^{3}$

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

         $$\int 10206 x\, dx = 10206 \int x\, dx$$

         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: $5103 x^{2}$

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

         $$\int 2187\, dx = 2187 x$$

      The result is: $16 x^{8} + 192 x^{7} + 1008 x^{6} + 3024 x^{5} + 5670 x^{4} + 6804 x^{3} + 5103 x^{2} + 2187 x$

2. Now simplify:

   $$\frac{\left(2 x + 3\right)^{8}}{16}$$

3. Add the constant of integration:

   $$\frac{\left(2 x + 3\right)^{8}}{16}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(2 x + 3\right)^{8}}{16}+ \mathrm{constant}$$