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.

If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$
intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$