The following is the reposted version of Kartik Sharma's problem on logarithmo-trigonometric integral:
Here is the solution to the problem:
In this case..first we observe that the nested radicals are of the form: which suggests that can be considered as for some ..Thus the nested radical part of the integral reduces to where We choose to be such that . Now, let . We use the two following results: and which are trivial.
Then we can write (clearly in steps 2 and 3 we have used Fubini's theorem and in step 2 I have used gamma function since .)
We can apply analytic continuation to the above result to extend the domain of from to since the original domain is open in and the integrand function is analytic in . Now we simply put to get that .