Integral of (4/x^2x+3x^2) dx

1. Integrate term-by-term:

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

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

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

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

      So, the result is: $x^{3}$

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

      $$\int \frac{4 x}{x^{2}}\, dx = 4 \int \frac{x}{x^{2}}\, dx$$

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

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

         1. Let $u = x^{2}$.

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

            $$\int \frac{1}{2 u}\, du$$

            1. Don't know the steps in finding this integral.

               But the integral is

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

            Now substitute $u$ back in:

            $$\log{\left(x^{2} \right)}$$

         So, the result is: $\frac{\log{\left(x^{2} \right)}}{2}$

      So, the result is: $2 \log{\left(x^{2} \right)}$

   The result is: $x^{3} + 2 \log{\left(x^{2} \right)}$

2. Add the constant of integration:

   $$x^{3} + 2 \log{\left(x^{2} \right)}+ \mathrm{constant}$$

The answer is:

$$x^{3} + 2 \log{\left(x^{2} \right)}+ \mathrm{constant}$$