I don't know latex but I have my work written out in . The answer seems to be incorrect and I am aware of the trig substitution trick that is usually used to solve this integral. I'm not interested in memorizing such a substitution and I don't need the answer since I can look it up if I need to. I'm wondering where my work went wrong here, and how to fix it so that memorizing dubious substitutions won't be necessary.
I also attempted this integral by writing sin(x) in exponential form via eulers identity and ended up with ln(abs((e^ix +1)/(e^ix -1 ))) +C as my answer, but here I am stuck trying to convert this back into trig terms. I suspect this answer is incorrect as well, probably for the same reason my first answer is incorrect.
What are non-memorization based ways of solving this integral and why is my approach incorrect?