Evaluate the definite integral:∫01(4x^3-2x^2+1)dx

Sure, here are the steps to evaluate the definite integral:

1. First, find the antiderivative of the function. The antiderivative of a function is found by reversing the process of differentiation.

   The antiderivative of 4x^3 is x^4, of -2x^2 is (2/3)x^3, and of 1 is x. So, the antiderivative of the function (4x^3 - 2x^2 + 1) is (x^4 - (2/3)x^3 + x).

2. Next, apply the Fundamental Theorem of Calculus, which states that the definite integral of a function from a to b is equal to the antiderivative at b minus the antiderivative at a.

   So, we evaluate the antiderivative at 1 and at 0 and subtract the two.

   At x = 1, the antiderivative is (1^4 - (2/3)1^3 + 1) = 1 - 2/3 + 1 = 4/3.

   At x = 0, the antiderivative is (0^4 - (2/3)0^3 + 0) = 0.

3. Subtract the two results: 4/3 - 0 = 4/3.

So, the definite integral of (4x^3 - 2x^2 + 1) from 0 to 1 is 4/3.