integrate sqrt(tan x)
I substituted tan x = u^2
then i got x = arc tan u^2
By substituting i got 2u^2/(1+u^4)
now what????
Now use partial fractions. Note that
and these two quadratics are irreducible. The ensuing algebra is your pleasant job to do.
Of interest: integrate Sqrt[Tan[x]] - Wolfram|Alpha
(Be sure to click on Show steps).