Let $${\psi _1}:[0,\infty ) \to R$$, $${\psi _2}:[0,\infty ) \to R$$, f : (0, $$\infty$$) $$\to$$ R and g : [0, $$\infty$$) $$\to$$ R be functions such that f(0) = g(0) = 0,

$${\psi _1}(x) = {e^{ - x}} + x,x \ge 0$$,

$${\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0$$,

$$f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x > 0} $$ and

$$g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x > 0} $$.

Which of the following statements is TRUE?

The value of $$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$ is equal to

Let $$f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$$ for all $$x \in R\,\,$$ with $$f\left( {{1 \over 2}} \right) = 0$$.
If $$m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $$ then the possible values of $$m$$ and $$M$$ are

Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt} $$ exists. Let this limit be $$g(a).$$ In addition, it is given that the function $$g(a)$$ is differentiable on $$(0,1).$$

The value of $$g'\left( {{1 \over 2}} \right)$$ is