The tangent to the curve, y = xe^x2 passing through the point (1, e) also passes through the point

The value of   $$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,$$  where [t] denotes the greatest integer less than or equal to t, is

Let S = $$\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.$$ Then S represents :

If  $$\int\limits_0^x \, $$f(t) dt = x² + $$\int\limits_x^1 \, $$ t²f(t) dt then f '$$\left( {{1 \over 2}} \right)$$ is -