So the integral question is:
I thought I would substitute in x=cos(u) dx=-sin(u)du ...
Wolfram Alpha uses x=sin(u)
But ultimately, I end up with cot(u), and wolfram gets tan(u) ... Last time I checked, these two are not the same function
What went wrong?
Either x= cos(u) or x= sin(u) will work because and and you get rid of that 1/2 power.
Yes, using x= cos(u) you get the integral as cot(u) and with x= sin(u) you get u= tan(x).
No, those are not the same function but since those are the results of different integrals, why would you expect them to be?
You haven't finished the problem yet!
If x= cos(u) then u= so, with the integral equal to tan(u), you get . If x= sin(u), then and you get .
Now, so . Of course, while . Your integral is .
so . Now and so the integral is , just as before.