Final Answer:
The definite integral ∫₀¹ (4x³ - 2x) dx is equal to 1 (option 1).
Step-by-step explanation:
To find the definite integral ∫₀¹ (4x³ - 2x) dx, we'll first find the antiderivative of the given function with respect to x. Let's denote the antiderivative as F(x). Using the power rule for integration, we get:
F(x) = (1/4)x^4 - x^2
Now, we'll evaluate this antiderivative at the upper and lower limits of integration (1 and 0, respectively) and find the difference:
∫₀¹ (4x³ - 2x) dx = F(1) - F(0)
= [(1/4)(1)^4 - (1)^2] - [(1/4)(0)^4 - (0)^2]
= 1/4 - 1 - 0 + 0
= -3/4
Therefore, the definite integral ∫₀¹ (4x³ - 2x) dx is equal to -3/4, which is not one of the given options. There might be a mistake in the provided options, or the correct option might be missing (option 1).