Final answer:
The antiderivative of f(x) = 2x - (1 - x)⁴ is calculated by integrating each term separately. The term (1 - x)⁴ requires a substitution for simplification, resulting in the final antiderivative x² - (1/5)(1 - x)⁵ + C, where C is the constant of integration.
Step-by-step explanation:
To find the antiderivative of the function f(x) = 2x - (1 - x)⁴, we need to integrate the function with respect to x. First, we recognize that the term (1 - x)⁴ is a composite function requiring a substitution to simplify the antiderivative process. Let's perform this step-by-step integration.
For the linear term 2x, the antiderivative is simply x². Now, considering the term (1 - x)⁴, let u = 1 - x. Therefore, -du/dx = 1, which implies that du = -dx, so we adjust the integral accordingly. The antiderivative of u⁴ is u⁵/5. We then substitute back for u to obtain (1 - x)⁵/5. Combining these parts gives us the complete antiderivative:
∫ f(x) dx = x² - ⅔(1 - x)⁵ + C
where C is the constant of integration.