$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$
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 functions defined implicitly by the equation $$y^3-3y+x=0$$ on various intervals in the real line. If $$x\in(-\infty,-2)\cup(2,\infty)$$, the equation implicitly defines a unique real valued differentiable function $$y=f(x)$$. If $$x\in(-2,2)$$, the equation implicitly defines a unique real valued differentiable function $$y=g(x)$$ satisfying $$g(0)=0$$
$$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$
$$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{\int\limits_2^{{{\sec }^2}x} {f(t)\,dt} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}$$ equal
Match the integrals in Column I with the values in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $$ | (P) | $${1 \over 2}\log \left( {{2 \over 3}} \right)$$ |
| (B) | $$\int\limits_0^1 {{{dx} \over {\sqrt {1 + {x^2}} }}} $$ | (Q) | $$2\log \left( {{2 \over 3}} \right)$$ |
| (C) | $$\int\limits_2^3 {{{dx} \over {1 + {x^2}}}} $$ | (R) | $${\pi \over 3}$$ |
| (D) | $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $$ | (S) | $${\pi \over 2}$$ |
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