Integral of (3x-1)(5x+)^8 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\left(5 x\right)^{8} \left(3 x - 1\right) = 1171875 x^{9} - 390625 x^{8}$$

   2. Integrate term-by-term:

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

         $$\int 1171875 x^{9}\, dx = 1171875 \int x^{9}\, dx$$

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

            $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

         So, the result is: $\frac{234375 x^{10}}{2}$

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

         $$\int \left(- 390625 x^{8}\right)\, dx = - 390625 \int x^{8}\, dx$$

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

            $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

         So, the result is: $- \frac{390625 x^{9}}{9}$

      The result is: $\frac{234375 x^{10}}{2} - \frac{390625 x^{9}}{9}$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 x\right)^{8} \left(3 x - 1\right) = - 390625 x^{8} + 3 x 390625 x^{8}$$

   2. Integrate term-by-term:

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

         $$\int \left(- 390625 x^{8}\right)\, dx = - 390625 \int x^{8}\, dx$$

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

            $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

         So, the result is: $- \frac{390625 x^{9}}{9}$

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

         $$\int 3 x 390625 x^{8}\, dx = 3 \int 390625 x x^{8}\, dx$$

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

            $$\int 390625 x x^{8}\, dx = 390625 \int x x^{8}\, dx$$

            1. Let $u = x^{2}$.

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

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

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

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

                  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: $\frac{u^{5}}{10}$

               Now substitute $u$ back in:

               $$\frac{x^{10}}{10}$$

            So, the result is: $\frac{78125 x^{10}}{2}$

         So, the result is: $\frac{234375 x^{10}}{2}$

      The result is: $\frac{234375 x^{10}}{2} - \frac{390625 x^{9}}{9}$

2. Now simplify:

   $$\frac{78125 x^{9} \left(27 x - 10\right)}{18}$$

3. Add the constant of integration:

   $$\frac{78125 x^{9} \left(27 x - 10\right)}{18}+ \mathrm{constant}$$

The answer is:

$$\frac{78125 x^{9} \left(27 x - 10\right)}{18}+ \mathrm{constant}$$