Integral of (x-4)*(x-6)/x^2 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\frac{\left(x - 6\right) \left(x - 4\right)}{x^{2}} = 1 - \frac{10}{x} + \frac{24}{x^{2}}$$

   2. Integrate term-by-term:

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

         $$\int 1\, dx = x$$

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

         $$\int \left(- \frac{10}{x}\right)\, dx = - 10 \int \frac{1}{x}\, dx$$

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

         So, the result is: $- 10 \log{\left(x \right)}$

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

         $$\int \frac{24}{x^{2}}\, dx = 24 \int \frac{1}{x^{2}}\, dx$$

         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: $- \frac{24}{x}$

      The result is: $x - 10 \log{\left(x \right)} - \frac{24}{x}$

   Method #2

   1. Rewrite the integrand:

      $$\frac{\left(x - 6\right) \left(x - 4\right)}{x^{2}} = \frac{x^{2} - 10 x + 24}{x^{2}}$$

   2. Rewrite the integrand:

      $$\frac{x^{2} - 10 x + 24}{x^{2}} = 1 - \frac{10}{x} + \frac{24}{x^{2}}$$

   3. Integrate term-by-term:

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

         $$\int 1\, dx = x$$

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

         $$\int \left(- \frac{10}{x}\right)\, dx = - 10 \int \frac{1}{x}\, dx$$

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

         So, the result is: $- 10 \log{\left(x \right)}$

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

         $$\int \frac{24}{x^{2}}\, dx = 24 \int \frac{1}{x^{2}}\, dx$$

         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: $- \frac{24}{x}$

      The result is: $x - 10 \log{\left(x \right)} - \frac{24}{x}$

2. Add the constant of integration:

   $$x - 10 \log{\left(x \right)} - \frac{24}{x}+ \mathrm{constant}$$

The answer is:

$$x - 10 \log{\left(x \right)} - \frac{24}{x}+ \mathrm{constant}$$