Integral of (4/x^2+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:

      $$\int \left(- 3 x^{2}\right)\, dx = - \int 3 x^{2}\, dx$$

      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: $x^{3}$

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

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

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

      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: $x^{2}$

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

      $$\int \frac{4}{x^{2}}\, dx = 4 \int \frac{1}{x^{2}}\, dx$$

      1. Rewrite the integrand:

         $$\frac{1}{x^{2}} = \tilde{\infty} 0$$

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

         $$\int \tilde{\infty} 0\, dx = \tilde{\infty} \int 0\, dx$$

         1. Integrate term-by-term:

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

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

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

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

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

            The result is: $0$

         So, the result is: $\text{NaN}$

      So, the result is: $\text{NaN}$

   The result is: $\text{NaN}$

2. Add the constant of integration:

   $$\text{NaN}+ \mathrm{constant}$$

The answer is:

$$\text{NaN}+ \mathrm{constant}$$