I was tutoring someone in Calculus today, and they had an extra credit assignment to do. These integrals appeared on past PUTNAM exams, and they can get help on these. However, I've been trying to work this one out and I (and all the other tutors I work with) couldn't figure it out.

So for the sake of me giving a hint or two to this person, I would like some guidance on how to tackle this beast:

All I know is that this equals 1 but I don't know how to get it. XD

Any input or suggestions would be appreciated!!