Integral of ∫(2x+6)^5 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 x + 6$.

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

      $$\int \frac{u^{5}}{2}\, du$$

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

         $$\int u^{5}\, du = \frac{\int u^{5}\, du}{2}$$

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

            $$\int u^{5}\, du = \frac{u^{6}}{6}$$

         So, the result is: $\frac{u^{6}}{12}$

      Now substitute $u$ back in:

      $$\frac{\left(2 x + 6\right)^{6}}{12}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(2 x + 6\right)^{5} = 32 x^{5} + 480 x^{4} + 2880 x^{3} + 8640 x^{2} + 12960 x + 7776$$

   2. Integrate term-by-term:

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

         $$\int 32 x^{5}\, dx = 32 \int x^{5}\, dx$$

         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: $\frac{16 x^{6}}{3}$

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

         $$\int 480 x^{4}\, dx = 480 \int x^{4}\, dx$$

         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: $96 x^{5}$

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

         $$\int 2880 x^{3}\, dx = 2880 \int x^{3}\, dx$$

         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: $720 x^{4}$

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

         $$\int 8640 x^{2}\, dx = 8640 \int x^{2}\, dx$$

         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: $2880 x^{3}$

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

         $$\int 12960 x\, dx = 12960 \int x\, dx$$

         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: $6480 x^{2}$

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

         $$\int 7776\, dx = 7776 x$$

      The result is: $\frac{16 x^{6}}{3} + 96 x^{5} + 720 x^{4} + 2880 x^{3} + 6480 x^{2} + 7776 x$

2. Now simplify:

   $$\frac{16 \left(x + 3\right)^{6}}{3}$$

3. Add the constant of integration:

   $$\frac{16 \left(x + 3\right)^{6}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{16 \left(x + 3\right)^{6}}{3}+ \mathrm{constant}$$