With a little of patience You can arrive to the general expression...
(1)
... and from (1) and the identity...
(2)
... You have finally...
(3)
Following a similar procedure You can solve also the second integral...
Kind regards
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.
Thanks.
I would just like to add the following for the OP with regards to applying this procedure to the second integral:
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 .