Integral of (5x-2/x^1/3) dx

1. Integrate term-by-term:

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

      $$\int 5 x\, dx = 5 \int x\, dx$$

      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: $\frac{5 x^{2}}{2}$

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

      $$\int \left(- \frac{2}{\sqrt[3]{x}}\right)\, dx = - 2 \int \frac{1}{\sqrt[3]{x}}\, dx$$

      1. Let $u = \sqrt[3]{x}$.

         Then let $du = \frac{dx}{3 x^{\frac{2}{3}}}$ and substitute $3 du$:

         $$\int 3 u\, du$$

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

            $$\int u\, du = 3 \int u\, du$$

            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: $\frac{3 u^{2}}{2}$

         Now substitute $u$ back in:

         $$\frac{3 x^{\frac{2}{3}}}{2}$$

      So, the result is: $- 3 x^{\frac{2}{3}}$

   The result is: $- 3 x^{\frac{2}{3}} + \frac{5 x^{2}}{2}$

2. Add the constant of integration:

   $$- 3 x^{\frac{2}{3}} + \frac{5 x^{2}}{2}+ \mathrm{constant}$$

The answer is:

$$- 3 x^{\frac{2}{3}} + \frac{5 x^{2}}{2}+ \mathrm{constant}$$