Let's step through the evaluation of this integral.
First, we should determine the integrand of this integral. Here, it's a polynomial, 1 + 2x^2.
Now, we find the antiderivative (also known as the indefinite integral) of the integrand. Recall that the antiderivative of a constant is just the constant times x and the antiderivative of x^n, where n is any real number, is x^(n+1)/(n+1). So, the antiderivative of 1 is x and the antiderivative of 2x^2 is 2/3 * x^3.
Summing these antiderivatives gives the antiderivative of the entire function: x + 4/3 * x^3.
Next, we'll use the Fundamental Theorem of Calculus, which says that the integral of a function from a to b equals the antiderivative evaluated at b minus the antiderivative evaluated at a. So, we'll plug 4 and 0 into x + 4/3 * x^3.
First plug 4 into x + 4/3 * x^3 to get 4 + 4/3 * 4^3.
Then, plug 0 into x + 4/3 * x^3, which just renders 0.
Finally, subtract the term you get from plugging in 0 from the term you get from plugging in 4. This is the calculation of the definite integral we want.
By evaluating the above, we find that the antiderivative of the given function is 4x^3/3 + x, and the definite integral evaluated from 0 to 4 is 268/3.
Therefore, ∫041+2x 2x dx = 268/3, is the final answer.