Integral of (5x+2)^3 dx

1. There are multiple ways to do this integral.

   Method #1

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

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{u^{3}}{5}\, du$$

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

         $$\int u^{3}\, du = \frac{\int u^{3}\, du}{5}$$

         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: $\frac{u^{4}}{20}$

      Now substitute $u$ back in:

      $$\frac{\left(5 x + 2\right)^{4}}{20}$$

   Method #2

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

         $$\int 125 x^{3}\, dx = 125 \int x^{3}\, dx$$

         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: $\frac{125 x^{4}}{4}$

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

         $$\int 150 x^{2}\, dx = 150 \int x^{2}\, dx$$

         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: $50 x^{3}$

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

         $$\int 60 x\, dx = 60 \int x\, dx$$

         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: $30 x^{2}$

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

         $$\int 8\, dx = 8 x$$

      The result is: $\frac{125 x^{4}}{4} + 50 x^{3} + 30 x^{2} + 8 x$

2. Now simplify:

   $$\frac{\left(5 x + 2\right)^{4}}{20}$$

3. Add the constant of integration:

   $$\frac{\left(5 x + 2\right)^{4}}{20}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(5 x + 2\right)^{4}}{20}+ \mathrm{constant}$$