This problem really made me depressed..
Heres what I did:
Then, I apply the half hangle formula..
This integrates to..
Now, got rid of the 2theta with another crazy identity..
Drew an actual triangle, and converted theta to x:
Nowww, after all this work.. when I plug in the upper limit of integration it is undefined! Apparently it isnt in the natural domain of arcsin..
How would I go about solving this? Did I integrate this correctly? I considered using hyperbolic substitution (sinh). Would this work? Would all the identities I just used with sin work with sinh aswell? If so, how do I convert theta -> x using hyperbolic trig functions?