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Let f(x) = integral of [(x-1) / ((x+1) * sqrt(x³ + x² + x))] dx, with the condition f(1) = 2*pi/3. Find the value of f'(1).

1. 0
2. pi/3
3. pi/4
4. 2

Correct answer: 0

Solution

Since f(x) is defined as an indefinite integral of g(x) = (x-1)/((x+1)*sqrt(x³+x²+x)), by the Fundamental Theorem of Calculus f'(x) = g(x). At x=1 the numerator x-1=0, making f'(1)=0.

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