Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by
$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$

Let $$g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$$ Which of the following is true?

Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by
$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$

Which of the following is true?

Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by
$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$

Which of the following is true?

Consider the functions defined implicitly by the equation
$${y^3} - 3y + x = 0$$ on various intervals in the real line. If
$$x \in \left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right),$$ the equation implicitly defines a unique
real valued differentiable function $$y = f\left( x \right).$$ If $$x \in \left( { - 2,2} \right),$$ the
equation implicitly defines a unique real valued differentiable function
$$y=g(x)$$ satisfying $$g(0)=0.$$

$$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$