Integral of (5x-3/2)^9 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 5 x - \frac{3}{2}$.

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

      $$\int \frac{u^{9}}{5}\, du$$

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

         $$\int u^{9}\, du = \frac{\int u^{9}\, du}{5}$$

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

            $$\int u^{9}\, du = \frac{u^{10}}{10}$$

         So, the result is: $\frac{u^{10}}{50}$

      Now substitute $u$ back in:

      $$\frac{\left(5 x - \frac{3}{2}\right)^{10}}{50}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(5 x - \frac{3}{2}\right)^{9} = 1953125 x^{9} - \frac{10546875 x^{8}}{2} + 6328125 x^{7} - \frac{8859375 x^{6}}{2} + \frac{15946875 x^{5}}{8} - \frac{9568125 x^{4}}{16} + \frac{1913625 x^{3}}{16} - \frac{492075 x^{2}}{32} + \frac{295245 x}{256} - \frac{19683}{512}$$

   2. Integrate term-by-term:

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

         $$\int 1953125 x^{9}\, dx = 1953125 \int x^{9}\, dx$$

         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: $\frac{390625 x^{10}}{2}$

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

         $$\int \left(- \frac{10546875 x^{8}}{2}\right)\, dx = - \frac{10546875 \int x^{8}\, dx}{2}$$

         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: $- \frac{1171875 x^{9}}{2}$

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

         $$\int 6328125 x^{7}\, dx = 6328125 \int x^{7}\, dx$$

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

            $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

         So, the result is: $\frac{6328125 x^{8}}{8}$

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

         $$\int \left(- \frac{8859375 x^{6}}{2}\right)\, dx = - \frac{8859375 \int x^{6}\, dx}{2}$$

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

            $$\int x^{6}\, dx = \frac{x^{7}}{7}$$

         So, the result is: $- \frac{1265625 x^{7}}{2}$

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

         $$\int \frac{15946875 x^{5}}{8}\, dx = \frac{15946875 \int x^{5}\, dx}{8}$$

         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: $\frac{5315625 x^{6}}{16}$

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

         $$\int \left(- \frac{9568125 x^{4}}{16}\right)\, dx = - \frac{9568125 \int x^{4}\, dx}{16}$$

         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: $- \frac{1913625 x^{5}}{16}$

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

         $$\int \frac{1913625 x^{3}}{16}\, dx = \frac{1913625 \int x^{3}\, dx}{16}$$

         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: $\frac{1913625 x^{4}}{64}$

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

         $$\int \left(- \frac{492075 x^{2}}{32}\right)\, dx = - \frac{492075 \int x^{2}\, dx}{32}$$

         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: $- \frac{164025 x^{3}}{32}$

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

         $$\int \frac{295245 x}{256}\, dx = \frac{295245 \int x\, dx}{256}$$

         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: $\frac{295245 x^{2}}{512}$

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

         $$\int \left(- \frac{19683}{512}\right)\, dx = - \frac{19683 x}{512}$$

      The result is: $\frac{390625 x^{10}}{2} - \frac{1171875 x^{9}}{2} + \frac{6328125 x^{8}}{8} - \frac{1265625 x^{7}}{2} + \frac{5315625 x^{6}}{16} - \frac{1913625 x^{5}}{16} + \frac{1913625 x^{4}}{64} - \frac{164025 x^{3}}{32} + \frac{295245 x^{2}}{512} - \frac{19683 x}{512}$

2. Now simplify:

   $$\frac{\left(10 x - 3\right)^{10}}{51200}$$

3. Add the constant of integration:

   $$\frac{\left(10 x - 3\right)^{10}}{51200}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(10 x - 3\right)^{10}}{51200}+ \mathrm{constant}$$