Originally Posted by
mathDad Hi (slow response on my part, sorry). Thanks for all your replies.
Yes, we had to use integration by parts. Integral of cos^2 was actually part of a larger trig substitution problem (which I'm not going to type up right now) where the teacher wanted them to "unsubstitute" using a triangle, and we couldn't figure out how to do that using the identity of pickslides response.
In other words, the whole problem was a trig substitution where we had to substitute (something like) x=3tan(t), and use use an identity to simplify sqrt(9-x^2) in the denominator and something else, blah blah. We resolved all the integrals, but were left with integral cos^2(t). At the end, the teacher wants them to unsubstitute by using tan(t)=x/3 and labeling the triangle's sides and figuring out how to "unsubstitute" (return to an equation in variable x). But the first trig identity leaves us with a cos(2t), and we do not see a way "unsubstitute" a trig function(2t) when all we have is tan(t)=x/3. I insisted on integration by parts because I wanted to keep the trig functions in t, rather than 2t.