Nice problems!

(1) can be written as

The first integral is known and its value is . The second can be calculated with contour integration, yielding .

(3) note that

and since for , we can put to obtain:

Also note that and .

From all these we obtain:

and when is a natural number, this is

(2) Let , with .

Then , and we're looking for this integral, evaluated when , .

To calculate , note that:

So that:

Now let .

It is easily checked that the indefinite integral with respect to of the right hand side of the previous equation is , so:

From the definition of it is obvious that the left hand side vanishes when , so the right hand side also vanishes. Thus, using , we get . Therefore:

Now we differentiate with respect to , remembering that :

.

Now, as we noted, , so finally we have:

.

Fortunately, when and the second term vanishes, and thus