Integral of (1/2x+1)⁴ dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \frac{x}{2} + 1$.

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

      $$\int 2 u^{4}\, du$$

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

         $$\int u^{4}\, du = 2 \int u^{4}\, du$$

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

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

         So, the result is: $\frac{2 u^{5}}{5}$

      Now substitute $u$ back in:

      $$\frac{2 \left(\frac{x}{2} + 1\right)^{5}}{5}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(\frac{x}{2} + 1\right)^{4} = \frac{x^{4}}{16} + \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + 2 x + 1$$

   2. Integrate term-by-term:

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

         $$\int \frac{x^{4}}{16}\, dx = \frac{\int x^{4}\, dx}{16}$$

         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: $\frac{x^{5}}{80}$

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

         $$\int \frac{x^{3}}{2}\, dx = \frac{\int x^{3}\, dx}{2}$$

         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: $\frac{x^{4}}{8}$

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

         $$\int \frac{3 x^{2}}{2}\, dx = \frac{3 \int x^{2}\, dx}{2}$$

         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: $\frac{x^{3}}{2}$

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

         $$\int 2 x\, dx = 2 \int x\, dx$$

         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: $x^{2}$

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

         $$\int 1\, dx = x$$

      The result is: $\frac{x^{5}}{80} + \frac{x^{4}}{8} + \frac{x^{3}}{2} + x^{2} + x$

2. Now simplify:

   $$\frac{\left(x + 2\right)^{5}}{80}$$

3. Add the constant of integration:

   $$\frac{\left(x + 2\right)^{5}}{80}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x + 2\right)^{5}}{80}+ \mathrm{constant}$$