Integral of (2x²-√x⁵+5)/x² 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 $- du$:

      $$\int \left(- \frac{2 u^{5} - 4 u^{4} - 10}{u^{3}}\right)\, du$$

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

         $$\int \frac{2 u^{5} - 4 u^{4} - 10}{u^{3}}\, du = - \int \frac{2 u^{5} - 4 u^{4} - 10}{u^{3}}\, du$$

         1. Rewrite the integrand:

            $$\frac{2 u^{5} - 4 u^{4} - 10}{u^{3}} = 2 u^{2} - 4 u - \frac{10}{u^{3}}$$

         2. Integrate term-by-term:

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

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

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

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

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

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

               $$\int \left(- 4 u\right)\, du = - 4 \int u\, du$$

               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: $- 2 u^{2}$

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

               $$\int \left(- \frac{10}{u^{3}}\right)\, du = - 10 \int \frac{1}{u^{3}}\, du$$

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

                  $$\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$$

               So, the result is: $\frac{5}{u^{2}}$

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

         So, the result is: $- \frac{2 u^{3}}{3} + 2 u^{2} - \frac{5}{u^{2}}$

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

         $$\int \left(- \sqrt{x}\right)\, dx = - \int \sqrt{x}\, dx$$

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

            $$\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$$

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

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

         $$\int 2\, dx = 2 x$$

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

         $$\int \frac{5}{x^{2}}\, dx = 5 \int \frac{1}{x^{2}}\, dx$$

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

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         So, the result is: $- \frac{5}{x}$

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

2. Add the constant of integration:

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

The answer is:

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