Let $$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.$$

Where a is a positive constant. Find the interval in which $$f'(x)$$ is increasing.

Evaluate $$\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $$.

Two guns, situated on the top of a hill of height 10 m, fire one shot each with the same speed $$5\sqrt 3 $$ m s^-1 at some interval of time. One gun fires horizontally and other fires upwards at an angle of $$60^\circ $$ with the horizontal. The shots collide in air at a point P. Find

(i) the time interval between the firings, and

(ii) the coordinates of the point P.

Take origin of the coordinate system at the foot of the hill right below the muzzle and trajectories in x-y plane.