If you try arccos of both of them, the + one gives pi/10, whereas the - one give 3pi/10, so I assume only the first one is correct as pi/10 = 18 degrees, maybe?
That question stumped me as well.
I think what they're saying is cos(18) ≈ 1, so cos (theta) must be as large as possible, hence you choose the + as it gives a larger number.
I'm so confused by that as well, when you use by parts it just ends up looping round and I have no idea what you're supposed to do
Edit: I think I just sorted it, if you write out the whole integration by parts formula, on the LHS you have the original integral, and on RHS you have the negative version of it, so add it onto the LHS, then you have 2(the original integral) so divide both sides by two, and then continue evaluating?