Consider the statement :
‘‘For an integer n, if n³ – 1 is even, then n is odd.’’
The contrapositive statement of this statement is :

For all twice differentiable functions f : R $$ \to $$ R,
with f(0) = f(1) = f'(0) = 0

If the tangent to the curve, y = f (x) = xlog_e x,
(x > 0) at a point (c, f(c)) is parallel to the line-segment
joining the points (1, 0) and (e, e), then c is equal to :

The integral $$\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$$ equals :