Integral of (5x-3)(sqrt)^3xdx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sqrt{x}$.

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

      $$\int \left(10 u^{8} - 6 u^{6}\right)\, du$$

      1. Integrate term-by-term:

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

            $$\int 10 u^{8}\, du = 10 \int u^{8}\, du$$

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

               $$\int u^{8}\, du = \frac{u^{9}}{9}$$

            So, the result is: $\frac{10 u^{9}}{9}$

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

            $$\int \left(- 6 u^{6}\right)\, du = - 6 \int u^{6}\, du$$

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

               $$\int u^{6}\, du = \frac{u^{7}}{7}$$

            So, the result is: $- \frac{6 u^{7}}{7}$

         The result is: $\frac{10 u^{9}}{9} - \frac{6 u^{7}}{7}$

      Now substitute $u$ back in:

      $$\frac{10 x^{\frac{9}{2}}}{9} - \frac{6 x^{\frac{7}{2}}}{7}$$

   Method #2

   1. Rewrite the integrand:

      $$x \left(5 x - 3\right) \left(\sqrt{x}\right)^{3} = 5 x^{\frac{7}{2}} - 3 x^{\frac{5}{2}}$$

   2. Integrate term-by-term:

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

         $$\int 5 x^{\frac{7}{2}}\, dx = 5 \int x^{\frac{7}{2}}\, dx$$

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

            $$\int x^{\frac{7}{2}}\, dx = \frac{2 x^{\frac{9}{2}}}{9}$$

         So, the result is: $\frac{10 x^{\frac{9}{2}}}{9}$

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

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

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

            $$\int x^{\frac{5}{2}}\, dx = \frac{2 x^{\frac{7}{2}}}{7}$$

         So, the result is: $- \frac{6 x^{\frac{7}{2}}}{7}$

      The result is: $\frac{10 x^{\frac{9}{2}}}{9} - \frac{6 x^{\frac{7}{2}}}{7}$

2. Now simplify:

   $$\frac{2 x^{\frac{7}{2}} \left(35 x - 27\right)}{63}$$

3. Add the constant of integration:

   $$\frac{2 x^{\frac{7}{2}} \left(35 x - 27\right)}{63}+ \mathrm{constant}$$

The answer is:

$$\frac{2 x^{\frac{7}{2}} \left(35 x - 27\right)}{63}+ \mathrm{constant}$$