∫√tan x

Let tanx=t²,..................(1)

then differentiating both sides,we get,

sec²x dx=2t dt

(1+tan²x) dx=2t dt

(1+t⁴) dx=2t dt.................from(1)

dx=[2t/(1+t⁴)] dt

now,

∫√tanx

=∫√(t²).[2t/(1+t⁴)] dt

=∫[2t²/(1+t⁴)] dt

and solve further.