Integral of x/x+8(2x-5)^3 dx

1. Integrate term-by-term:

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

      $$\int 8 \left(2 x - 5\right)^{3}\, dx = 8 \int \left(2 x - 5\right)^{3}\, dx$$

      1. There are multiple ways to do this integral.

         Method #1

         1. Let $u = 2 x - 5$.

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

            $$\int \frac{u^{3}}{4}\, du$$

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

               $$\int \frac{u^{3}}{2}\, du = \frac{\int u^{3}\, du}{2}$$

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

                  $$\int u^{3}\, du = \frac{u^{4}}{4}$$

               So, the result is: $\frac{u^{4}}{8}$

            Now substitute $u$ back in:

            $$\frac{\left(2 x - 5\right)^{4}}{8}$$

         Method #2

         1. Rewrite the integrand:

            $$\left(2 x - 5\right)^{3} = 8 x^{3} - 60 x^{2} + 150 x - 125$$

         2. Integrate term-by-term:

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

               $$\int 8 x^{3}\, dx = 8 \int x^{3}\, dx$$

               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: $2 x^{4}$

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

               $$\int \left(- 60 x^{2}\right)\, dx = - 60 \int x^{2}\, dx$$

               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: $- 20 x^{3}$

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

               $$\int 150 x\, dx = 150 \int x\, dx$$

               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: $75 x^{2}$

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

               $$\int \left(-125\right)\, dx = - 125 x$$

            The result is: $2 x^{4} - 20 x^{3} + 75 x^{2} - 125 x$

      So, the result is: $\left(2 x - 5\right)^{4}$

   1. Don't know the steps in finding this integral.

      But the integral is

      $$x$$

   The result is: $x + \left(2 x - 5\right)^{4}$

2. Now simplify:

   $$x + \left(2 x - 5\right)^{4}$$

3. Add the constant of integration:

   $$x + \left(2 x - 5\right)^{4}+ \mathrm{constant}$$

The answer is:

$$x + \left(2 x - 5\right)^{4}+ \mathrm{constant}$$