Integral of (5x+2)^5dx 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 $\frac{du}{5}$:

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

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

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

         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: $\frac{u^{6}}{30}$

      Now substitute $u$ back in:

      $$\frac{\left(5 x + 2\right)^{6}}{30}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 x + 2\right)^{5} = 3125 x^{5} + 6250 x^{4} + 5000 x^{3} + 2000 x^{2} + 400 x + 32$$

   2. Integrate term-by-term:

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

         $$\int 3125 x^{5}\, dx = 3125 \int x^{5}\, dx$$

         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: $\frac{3125 x^{6}}{6}$

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

         $$\int 6250 x^{4}\, dx = 6250 \int x^{4}\, dx$$

         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: $1250 x^{5}$

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

         $$\int 5000 x^{3}\, dx = 5000 \int x^{3}\, dx$$

         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: $1250 x^{4}$

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

         $$\int 2000 x^{2}\, dx = 2000 \int x^{2}\, dx$$

         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: $\frac{2000 x^{3}}{3}$

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

         $$\int 400 x\, dx = 400 \int x\, dx$$

         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: $200 x^{2}$

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

         $$\int 32\, dx = 32 x$$

      The result is: $\frac{3125 x^{6}}{6} + 1250 x^{5} + 1250 x^{4} + \frac{2000 x^{3}}{3} + 200 x^{2} + 32 x$

2. Now simplify:

   $$\frac{\left(5 x + 2\right)^{6}}{30}$$

3. Add the constant of integration:

   $$\frac{\left(5 x + 2\right)^{6}}{30}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(5 x + 2\right)^{6}}{30}+ \mathrm{constant}$$