Sketch the region bounded by the curves $$y = {x^2}$$ and
$$y = {2 \over {1 + {x^2}}}.$$ Find the area.

Determine a positive integer $$n \le 5,$$ such that $$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$$

Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$

Sketch the curves and identify the region bounded by
$$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.