Don't hide your solutions.
1)
2)
3)
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