I've been chipping away at these two problems for a couple days now - they're part of a much larger assignment that is due today - and I've had no luck whatsoever with them. Every time I think a lightbulb has lit up, I end up with a couple pages of work proving nothing. Here are the two questions, and if anybody has any suggestions they would be greatly appreciated.
1) I = (integral) x*(arctanx)^2dx
For this one I'm supposed to solve it in terms of x, but I've tried integrating by parts, using x=([x^2]/2)', and then in the resulting equation substituting tanu for x in the integral.
It took me from (integral) [(arctanx)x^2]/(1+x^2) dx to (integral)u(tanu)^2du but I think that restricts u to being on +/- pi/2. Either way, I didn't get any useful answer out of it.
2) I = (integral) [(x^2 + 1)^n]dx
I have to find a reduction formula for this question, but I've tried integrating by parts and trig substitution to no avail. When I integrate by parts, I just get an x with an increasing power for every repetition in the integral, and trig substitution has just been a disaster so far.
If anybody knows how I could do these without killing another forest for paper that would be great.