I have made some further attempts to solve this.
It seems like the integrands in problem 1) are actually related as
Could it be that the definition of what is meant by that I have used is wrong? What is the most common definition for this? In the book I am studying, the definition is only given for sequences. It is stated that for two sequences and we have the definition iff or equivalently . Perhaps this is not generalized to functions in the way I thought.
Alternatively, there is some other trick that should be made in order to solve problems 1) and 2), or perhaps (but probably very unlikely), the problems are wrong.
I would be very thankful for any help, suggestions or comments.