Integral of 3*x*dx/(sqrt(x)-2) dx

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

   Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $6 du$:

   $$\int \frac{6 u^{3}}{u - 2}\, du$$

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

      $$\int \frac{u^{3}}{u - 2}\, du = 6 \int \frac{u^{3}}{u - 2}\, du$$

      1. Rewrite the integrand:

         $$\frac{u^{3}}{u - 2} = u^{2} + 2 u + 4 + \frac{8}{u - 2}$$

      2. Integrate term-by-term:

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

            $$\int u^{2}\, du = \frac{u^{3}}{3}$$

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

            $$\int 2 u\, du = 2 \int u\, du$$

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

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

            So, the result is: $u^{2}$

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

            $$\int 4\, du = 4 u$$

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

            $$\int \frac{8}{u - 2}\, du = 8 \int \frac{1}{u - 2}\, du$$

            1. Let $u = u - 2$.

               Then let $du = du$ and substitute $du$:

               $$\int \frac{1}{u}\, du$$

               1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

               Now substitute $u$ back in:

               $$\log{\left(u - 2 \right)}$$

            So, the result is: $8 \log{\left(u - 2 \right)}$

         The result is: $\frac{u^{3}}{3} + u^{2} + 4 u + 8 \log{\left(u - 2 \right)}$

      So, the result is: $2 u^{3} + 6 u^{2} + 24 u + 48 \log{\left(u - 2 \right)}$

   Now substitute $u$ back in:

   $$2 x^{\frac{3}{2}} + 24 \sqrt{x} + 6 x + 48 \log{\left(\sqrt{x} - 2 \right)}$$

2. Add the constant of integration:

   $$2 x^{\frac{3}{2}} + 24 \sqrt{x} + 6 x + 48 \log{\left(\sqrt{x} - 2 \right)}+ \mathrm{constant}$$

The answer is:

$$2 x^{\frac{3}{2}} + 24 \sqrt{x} + 6 x + 48 \log{\left(\sqrt{x} - 2 \right)}+ \mathrm{constant}$$