With a little of patience You can arrive to the general expression...
... and from (1) and the identity...
... You have finally...
Following a similar procedure You can solve also the second integral...
I'm trying to get the exact value of following definite integrals without using a CAS or tables. Any help would be appreciated:
I can get answers using the Wolframalpha website and also from a table of integrals. But I'd really like a calculus approach. I can't see what contour to use using contour integration. I can't see how to 'differentiate under the integral'. I can't see any substitution that will help. I know they can be done otherwise there would be no answer in the tables. I'm stuck.
I've seen a few members who are absolutely amazing in the sorts of integrals they have solved. Hopefully someone can help me with these ones.
It cannot be used directly because the series for only converges for |x| < 1.
So I suggest that as part of your attempt at a solution you consider and make the substitution .