19/09/07 21:50:08.96 K5YuQxfQ.net
f(1) = f(1) - f(0) = ∫_{0}^{1} f'(x) dx = ∫_{0}^{1} 1 / (1 + x + x^2) dx
= ∫_{0}^{1} 1 / [(x + 1/2)^2 + 3/4] dx
= ∫_{0}^{1} (4/3) / [((2/sqrt(3))*x + 1/sqrt(3))^2 + 1] dx
= (4/3) * ∫_{0}^{1} 1 / [((2/sqrt(3))*x + 1/sqrt(3))^2 + 1] dx
= (4/3) * ∫_{1/sqrt(3)}^{sqrt(3)} (1 / (t^2 + 1)) * sqrt(3)/2 dt
= (4/3) * sqrt(3)/2 * ∫_{1/sqrt(3)}^{sqrt(3)} (1 / (t^2 + 1)) dt
= (4/3) * sqrt(3)/2 * [arctan(sqrt(3)) - arctan(1/sqrt(3))]
= (2/sqrt(3)) * [π/3 - π/6]
= (2/sqrt(3)) * π/6
= π/(3*sqrt(3))