Integral of (10x^4+3x^2+4) dx

1. Integrate term-by-term:

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

      $$\int 10 x^{4}\, dx = 10 \int x^{4}\, dx$$

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

         $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

      So, the result is: $2 x^{5}$

   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 is the constant times the variable of integration:

      $$\int 4\, dx = 4 x$$

   The result is: $2 x^{5} + x^{3} + 4 x$

2. Now simplify:

   $$x \left(2 x^{4} + x^{2} + 4\right)$$

3. Add the constant of integration:

   $$x \left(2 x^{4} + x^{2} + 4\right)+ \mathrm{constant}$$

The answer is:

$$x \left(2 x^{4} + x^{2} + 4\right)+ \mathrm{constant}$$