Integral of (4f(x)+3g(x))dx dx

1. Rewrite the integrand:

   $$\left(4 f x + 3 g x\right) 1 = x \left(4 f + 3 g\right)$$

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

   $$\int x \left(4 f + 3 g\right)\, dx = \left(4 f + 3 g\right) \int x\, dx$$

   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: $\frac{x^{2} \cdot \left(4 f + 3 g\right)}{2}$

3. Add the constant of integration:

   $$\frac{x^{2} \cdot \left(4 f + 3 g\right)}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \cdot \left(4 f + 3 g\right)}{2}+ \mathrm{constant}$$