I am very interested in problems of integration and limit determination. In order to be able to continue other work with the help of Maple, I would like some advice on solving the following problem. The solution I know of using pen and paper is very tiring, perhaps Maple can make it easier:

To calculate Int(from 0 to 1)[ln(x)*ln(1-x)]dx
or, for obvious reasons, formulated differently:
Let eps, delta>0. Calculate
lim(eps, delta-->0)Int(0+eps to 1-delta)[ln(x)*ln(1-x)]dx

One of the most difficult/nice tasks from my collection of old problems:
Calculate Int(from 0 to 1)Int(from 0 to 1)[1/(1-x*y)]dxdy .

Let eps be a real number greater than zero. Calculate the limit lim(x-->oo)[x^(1-eps)*Integral(from x to x+1) sin(t^2)dt] .

The square grid is given in the Euclidean plane. All grid points therefore have integer coordinates. Investigate whether there are three points in this grid plane that form an equilateral triangle.

A regular polygon with n sides of length 1 is drawn. Then, the n midpoints of the edges are used to create another regular polygon inside the first. What is the length of an edge of the new polygon? As n approaches infinity, what does this length approach?

Hopefully this task is suitable for Maple?