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I worked this problem for a quite a while last night. I posted it here.

https://math.stackexchange.com/questions/3547225/help-with-trig-sub-integral/3547229#3547229

The original problem is in the top left. Sorry that the negative sometimes gets cut off in the photo, and yes I know it's not fully simplified there.

My first question is the more involved one: Is the algebra in my original work sound? If it is, why doesn't it work in this instance?

My second question is: is this a correct solution? https://imgur.com/2tgEz0O

It is for webwork, and I used two out of three chances. I'd prefer to keep my perfect webwork mark, obviously ;p

Finally, I was kind of impressed with Ans4's square completion and had to run it through to see that it was correct. That's such a useful skill. Do you have some advice about how I could improve my math tricks to that point?