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Consider a function f defined on the interval [0, 2] such that it is continuous on [0, 2] and differentiable on (0, 2), with f(0) = 1. Define F(x) = ∫₀ˣ f(√t) dt for x ∈ [0, 2]. If it is given that F'(x) = f'(x) for every x in (0, 2), what is the value of F(2)?

1. e² - 1
2. e⁴ - 1
3. e - 1
4. e⁴

Correct answer: e⁴ - 1

Solution

Using the given conditions and applying the Fundamental Theorem of Calculus, F'(x) = f'(x) implies that F(x) = f(x) + C. Evaluating F(2) with the given function leads to the result e⁴ - 1.

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