I have: 64 integral from (pi/4) to (pi/3) (1+2cos(2theta)+cos(2theta)^2) dtheta. Then I split those up into the integrals (still multiplying everything by 64) of 1, 2cos(2theta) and cos(2theta)^2. I have a lot of work shown, but I ended up getting the answer 64(3pi/32+sin(pi/6)+1/8sin(pi/3). That answer isn't right and I simplified it to the best of my ability and I don't know where I went wrong! But for the integral of 1, I got (pi/3-pi/4) and for the integral of 2cos(2theta) I did u substitution and got sin(2theta) and used the limits, and then for cos(2theta)^2 I used the half angle formula and got (1+cos(4theta) over 2, since its double that of a single half angle formula. Using the formula, I got (1/8) integral of 1 + integral sin(4theta). SO that was for the last integral when I first split everything up.
Well (1) / (1/4)^4 equals 256, and I got 64 because after multiplying the 2 half angle formulas of cosine, I took out that (1/4) and multiplied 256 by it. After multiplying (1+cos(2theta)/2 by itself, you get (1+2cos(2theta)+cos(2theta)^2 over 4. I took out that 4 (in terms of 1/4) and multiplied 256 by it. I don't know if that makes sense. I don't know what latex is! I'm sorry!
Well (radical 2 all over 4) and (1/2) are the bounds from the original problem, and I used cos(30degrees) and cos(45degrees) to convert to theta, giving me (pi/3) and (pi/4). So for (1/2), I set that equal to (1/4)sectheta, then divided (1/2) by (1/4) and got 2. Then I set 2=(1)/(costheta) since that is the same thing as sectheta, and then got costheta=1/2. Then I got the angle 60 I think and that got me pi over 3. Thats how I did the other one too! Maybe that was wrong?
Yep, you're right!! To tell you the truth, I think I messed up the integral of 1!! That's embarrassing. I think that's where I went wrong, because everything else looks the same, except for the 3theta over 2. I had 3theta over 32, which I got from adding (pi/12) to (pi/96). Yeah I did that wrong. Thank you SO much, though for taking the time!!! Now I have to study this so I get it right next time!!