Please see Red
Thanks!
-qbkr21
Hmmm...for some odd reason I though you integrated first with "u" before you substitute "u" back into the problem, is this not always the case?
Ohh...I see I just substituted for cos(x) not 1+cos^2(x). I think that I made this a bit more difficult than it had to be.
-qbkr21
the whole point of substitution is to turn a complicated integral into a simple one. the simpler we can make things the better. i realized that substituting u = cosx or u = cos^2x only would not really get rid of the problem, since i'd have to make a second substitution, say v = u^2 to get rid of the resulting u in the numerator, or would have to deal with a 1 + u in the denominator (not really a big deal, but still). now if i substituted 1 + cos^2x, i would get rid of the constant and the cos^2x, the derivative of a constant is zero, so it doesn't matter in the long run. and the integral will simply be -1/u, and you can't get any better than that
looking at the problem again, i saw that you just substituted cosx, but as you see, you can make life a lot simpler than that
Unless you are required to use a particular method when confronted with
the integral:
I = 2* integral u/(1+u^2) du
the observation that 2u = d/du[1+u^2] should imeadiatly tell you that:
I = ln(1+u^2) + c.
The general idea is that:
integral [df/dx]/f(x) dx = ln(f(x) + c
RonL